Examples in Continuum Theory

Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih November 29, 2001

2

Contents

0 PRELIMINARIES 7

1 LOCALLY CONNECTED CONTINUA 9 1.1 Basic examples: an arc, a circle, n-cube 𝕀^{n}, Hilbert cube …. 9 1.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Dendrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Preliminary properties . . . . . . . . . . . . . . . . . . . 10 1.3.2 Simple examples . . . . . . . . . . . . . . . . . . . . . . 11 1.3.3 The locally connected fan . . . . . . . . . . . . . . . . . 11 1.3.4 The locally connected combs . . . . . . . . . . . . . . . 11 1.3.5 Universal dendrites . . . . . . . . . . . . . . . . . . . . . 12 1.3.6 Ważewski universal dendrite . . . . . . . . . . . . . . . . 12 1.3.7 Universal dendrites of order m . . . . . . . . . . . . . . 13 1.3.8 Other universal dendrites . . . . . . . . . . . . . . . . . 17 1.3.9 The dendrite P . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.10 Self-homeomorphic dendrites . . . . . . . . . . . . . . . 18 1.3.11 Monotone equivalence and monotone homogeneity … 19 1.3.12 Omiljanowski dendrite . . . . . . . . . . . . . . . . . . . 19 1.3.13 Chaotic and rigid dendrites . . . . . . . . . . . . . . . . 23 1.3.14 Miller dendrite . . . . . . . . . . . . . . . . . . . . . . . 24 1.3.15 Dendrites of de Groot-Wille type . . . . . . . . . . . . . 24 1.3.16 Modified Miller dendrites . . . . . . . . . . . . . . . . . 26 1.3.17 Dendrites with the closed set of end points . . . . . . . . 26 1.3.18 Gehman dendrite . . . . . . . . . . . . . . . . . . . . . . 27 1.3.19 Modifications of the Gehman dendrite . . . . . . . . . . 27 1.3.20 Mapping hierarchy of dendrites . . . . . . . . . . . . . . 28

1.4 Cyclic examples of locally connected continua . . . . . . . . . . 29 1.4.1 Menger universal continua . . . . . . . . . . . . . . . . . 29 1.4.2 Sierpiński carpet . . . . . . . . . . . . . . . . . . . . . . 31 1.4.3 Menger universal curve . . . . . . . . . . . . . . . . . . . 34

1.5 2-dimensional locally connected continua . . . . . . . . . . . . . 36 1.6 Higher dimensional locally connected continua . . . . . . . . . . 36 1.7 Variations of local connectedness . . . . . . . . . . . . . . . . . . 36 2 JOINING PROPERTIES 37

2.1 Cut points and separating points (also local) . . . . . . . . . . . 37 2.2 Arcwise connectedness; arc components . . . . . . . . . . . . . . 37 2.3 Continuum chainability . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Various kinds of connectedness (λ-connectedness,

δ-connectedness, etc.) . . . . . . . . . . . . . . . . . . . . . . . . 37

3 DECOMPOSABLE AND INDECOMPOSABLE CONTINUA 39 3.1 Decomposable continua (also hereditary) . . . . . . . . . . . . . 39 3.2 Indecomposable continua (also hereditary) . . . . . . . . . . . . 39

3.2.1 Knaster continua . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Unicoherent continua (also hereditary) . . . . . . . . . . . . . . 43 3.4 Discoherent continua . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5 Multicoherent continua . . . . . . . . . . . . . . . . . . . . . . . 43

4 IRREDUCIBLE CONTINUA 45

4.1 Elementary examples . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.1 Sin curve . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.2 Cantor organ and accordion . . . . . . . . . . . . . . . . 47

4.2 Arc-like (chainable) continua . . . . . . . . . . . . . . . . . . . . 48 4.3 Irreducible circle-like continua . . . . . . . . . . . . . . . . . . . 48

4.3.1 Solenoids . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4 Compactifications of the real line and the real half-line . . . . . 51 4.5 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 USC DECOMPOSITIONS OF CONTINUA 53 5.1 Decomposition spaces . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 Upper semi-continuous decompositions . . . . . . . . . . . . . . 53 5.3 Continuous decompositions . . . . . . . . . . . . . . . . . . . . . 53

6 CURVES 55 6.1 Order of a space at a point . . . . . . . . . . . . . . . . . . . . . 55 6.2 Case-Chamberlin examples . . . . . . . . . . . . . . . . . . . . . 55 6.3 Curves without arcs . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.4 Covering defined classes (inverse limit expansions) . . . . . . . . 55 6.5 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7 FAMILIES OF CONTNINUA 57 7.1 Universal elements . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.2 Common models . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.3 Families of incomparable continua . . . . . . . . . . . . . . . . . 57

8 PLANE CONTINUA 59 8.1 Embeddability of continua in surfaces . . . . . . . . . . . . . . . 59 8.2 Planability in general . . . . . . . . . . . . . . . . . . . . . . . . 59 8.3 Planability of graphs on of locally connected continua . . . . . . 59

8.4 Separation of the plane . . . . . . . . . . . . . . . . . . . . . . . 59

9 MAPPINGS 61 9.1 Various kinds of mappings and relations between them . . . . . 61

10 CONTINUA WITH SPECIAL MAPPING PROPERTIES 63 10.1 Existence of special mappings . . . . . . . . . . . . . . . . . . . 63 10.2 Self-similar continua . . . . . . . . . . . . . . . . . . . . . . . . . 63 10.3 Self-homeomorphic continua . . . . . . . . . . . . . . . . . . . . 63 10.4 Rigid continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 10.5 Chaotic continua . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 10.6 Absolute retracts; spaces LC^{n} and LC^{∞} . . . . . . . . . . . . . 64 10.7 Absolute retracts for various classes of continua . . . . . . . . . 64 10.8 Class (W) and other classes defined by mappings . . . . . . . . . 64 10.9 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 10.10 Variations of homogeneity . . . . . . . . . . . . . . . . . . . . . 64 10.11 Generalized homogeneity . . . . . . . . . . . . . . . . . . . . . . 64 10.12 Embeddability in ℝ^{n} . . . . . . . . . . . . . . . . . . . . . . . . 64 10.13 Weakly chainable continua . . . . . . . . . . . . . . . . . . . . . 64 10.14 Fixed point property . . . . . . . . . . . . . . . . . . . . . . . . 64

11 CONTINUA AS FRACTALS 65

12 DYNAMICAL SYSTEMS 67 12.1 Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

13 OTHER PROPERTIES 69 13.1 Colocal connectedness . . . . . . . . . . . . . . . . . . . . . . . . 69 13.2 Aposyndesis and its variations . . . . . . . . . . . . . . . . . . . 69 13.3 Property of Kelley (also hereditary) . . . . . . . . . . . . . . . . 69 13.4 Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 13.5 Arc-smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 13.6 C^{∗}-smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

14 NEW FIGURES 71

14.1 New examples 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 14.1.1 Whyburn’s curve W . . . . . . . . . . . . . . . . . . . . 71 14.1.2 Pseudo-arc . . . . . . . . . . . . . . . . . . . . . . . . . 71 14.1.3 Zig-zag circle . . . . . . . . . . . . . . . . . . . . . . . . 71 14.1.4 M-continuum . . . . . . . . . . . . . . . . . . . . . . . . 71 14.1.5 Cantor interaction . . . . . . . . . . . . . . . . . . . . . 71 14.1.6 Pseudo-circle . . . . . . . . . . . . . . . . . . . . . . . . 71 14.1.7 Semicircle continuum . . . . . . . . . . . . . . . . . . . . 71 14.1.8 Boxes continuum . . . . . . . . . . . . . . . . . . . . . . 80

14.2 New examples 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 14.2.1 Graph K_{3,3} . . . . . . . . . . . . . . . . . . . . . . . . . 80 14.2.2 Graph K_{5} . . . . . . . . . . . . . . . . . . . . . . . . . . 80 14.2.3 Space filling curve . . . . . . . . . . . . . . . . . . . . . 81 14.2.4 Universal arc-like continuum . . . . . . . . . . . . . . . . 81 14.2.5 Lakes of Vada . . . . . . . . . . . . . . . . . . . . . . . . 81 14.2.6 Borsuk fan . . . . . . . . . . . . . . . . . . . . . . . . . . 81 14.2.7 Ingram’s atriodic continuum . . . . . . . . . . . . . . . . 84 14.2.8 Double Warsaw circle . . . . . . . . . . . . . . . . . . . 84 14.2.9 Warsaw circle . . . . . . . . . . . . . . . . . . . . . . . . 84 14.2.10 Crooked Warsaw circle . . . . . . . . . . . . . . . . . . . 84

14.3 New examples 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 14.3.1 Young spiral . . . . . . . . . . . . . . . . . . . . . . . . . 84 14.3.2 Young space . . . . . . . . . . . . . . . . . . . . . . . . . 84 14.3.3 Bing space . . . . . . . . . . . . . . . . . . . . . . . . . . 89 14.3.4 Universal smooth dendroid . . . . . . . . . . . . . . . . 89 14.3.5 Claytor’s curve A . . . . . . . . . . . . . . . . . . . . . . 89 14.3.6 Claytor’s curve B . . . . . . . . . . . . . . . . . . . . . . 89 14.3.7 Sin addiction continuum . . . . . . . . . . . . . . . . . . 89 14.3.8 Dumbbell . . . . . . . . . . . . . . . . . . . . . . . . . . 92 14.3.9 Figure Eight . . . . . . . . . . . . . . . . . . . . . . . . . 92 14.3.10 Noose . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

14.4 New examples 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 14.4.1 Theta curve . . . . . . . . . . . . . . . . . . . . . . . . . 93 14.4.2 Trapezoid basins continuum . . . . . . . . . . . . . . . . 93 14.4.3 Dragon continuum . . . . . . . . . . . . . . . . . . . . . 93

14.5 New examples 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 14.5.1 Harmonic fan . . . . . . . . . . . . . . . . . . . . . . . . 96 14.5.2 Harmonic shredded fan . . . . . . . . . . . . . . . . . . . 96 14.5.3 Cantor fan . . . . . . . . . . . . . . . . . . . . . . . . . . 96 14.5.4 Lelek fan . . . . . . . . . . . . . . . . . . . . . . . . . . 96

14.6 New examples 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 14.6.1 Sierpiński triangle . . . . . . . . . . . . . . . . . . . . . 98 14.6.2 Makuchowski umbrella . . . . . . . . . . . . . . . . . . . 98 14.6.3 Snake . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 14.6.4 Cantor snake . . . . . . . . . . . . . . . . . . . . . . . . 98 14.6.5 Cruller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

14.7 New examples 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 14.7.1 Cantor comb . . . . . . . . . . . . . . . . . . . . . . . . 98 14.7.2 Gehman dendrite on ski . . . . . . . . . . . . . . . . . . 98 14.7.3 Makuchowski twin . . . . . . . . . . . . . . . . . . . . . 102 14.7.4 Cantor function comb . . . . . . . . . . . . . . . . . . . 102 14.7.5 Oversteegen’s dendroid . . . . . . . . . . . . . . . . . . . 102

A Definitions 107

B List of examples 131

Chapter 0

PRELIMINARIES

All spaces considered here are assumed to be metric (if otherwise is not explicitly said) and all mappings are assumed to be continuous. Concepts and notions undefined in the text are used according to their meaning known from the literature. To aviod any doubt or misunderstanding, the reader can find the needed definition or explanation in Appendix A, where the items are alphabetically ordered.

The following notation will be used. The symbol ℝ stands for the space of real numbers, equipped with the natural topology. Thus ℝ^{2} denotes the plane (usually supplied with a Cartesian coordinate system); equivalently, the plane can be denoted by ℂ when considered as the set of all complex numbers. The closed unit interval [0, 1] of reals is denoted by 𝕀, and ℕ means the set of all positive integers.

8 CHAPTER 0. PRELIMINARIES

LOCALLY CONNECTED CONTINUA

1.1 Basic examples: an arc, a circle, n-cube 𝕀^{n}, Hilbert cube

1.2 Graphs

(a) Elementary examples: an arc, a circle, a noose, a figure eight, a θ-curve

(b) Acyclic graphs = trees.

(c) planability of graphs; the Kuratowski graphs K_{3,3}, K_{5}; Whyburn’s example of an open map from a planar onto a nonplanar graph

(d) Problem of Königsberg bridges; Euler’s solution

1.3 Dendrites

A dendrite is a locally connected continuum containing no simple closed curve. By the order of a point p in a dendrite X, writing ord (p, X)ord (p, X), we mean the Menger-Urysohn order, see [Kuratowski 1968, §51, I, p. 274], or equivalently, the order in the classical sense, i.e., the number of arcs emanating from p and disjoint out of p (see [Charatonik 1962, p. 229] and [Lelek 1961, p. 301]).

In this chapter the term of an end point of a continuum is always used in the sense of a point of (Menger-Urysohn) order 1, i.e., p is an end point of X provided that ord (p, X) = 1. For dendrites this concept coincides with one of an end point in the classical sense, but not with the notion of an end point of an arc-like continuum as defined e.g. in [Bing 1951, p. 660].

Given a dendrite X, we denote by E(X) the set of all end points of X and by R(X) the set of all its ramification points (i.e., points of order at least 3). Various structural as well as mapping characterizations of dendrites are collected in [Charatonik et al. 1998, Theorems 1.1 and 1.2, p. 228 and 230, respectively]. See also [Charatonik et al. 2000].

1.3.1 Preliminary properties

The following properties of dendrites are known (see e.g. [Nadler 1992, Chapter 10, p. 165], [Kuratowski 1968, §51, VI, p. 300] or [Whyburn 1942, p. 88]; compare also [Charatonik et al. 1998, Theorems 1.3 and 1.5, p. 233]).

1. Every subcontinuum of a dendrite is a dendrite (thus each dendrite is hereditarily locally connected).

2. Every connected subset of a dendrite is arcwise connected.

3. In any dendrite X the order of a point p is equal to the number of components of X \ {p}.

4. The set of all end points of a dendrite is 0-dimensional.

5. The set of all end points of a dendrite is a G_{δ}-set.

6. The set of all ordinary points (i.e., points of order 2) of a dendrite X is dense in X.

7. Every dendrite has at most countably many ramification points.

8. If m is the order of a ramification point in a dendrite, then m ∈ {3, 4, …, ω}.

9. No dendrite contains points of order ℵ_{0} or 𝔠 (i.e., each dendrite is a regular continuum).

10. In any dendrite the set of all its end points is dense if and only if the set of all its ramification points is dense.

11. Dendrites are absolute retracts. In fact, the class of all 1-dimensional absolute retracts coincide with the class of dendrites, [Borsuk 1967, Corollary 13.5, p. 138].

12. A continuum X is a dendrite if and only if for every compact space (continuum) Y and for every light confluent mapping f : Y → f(Y) such that X ⊂ f(Y) there is a copy X′ of X in Y for which the restriction f|X′ : X′ → X is a homeomorphism, [Charatonik et al. XXXXb].

Figure 1.3.3: ( A ) locally connected fan F_{ω}

1.3.2 Simple examples

The simplest examples of dendrites are the arc and n-ods for n ∈ {3, 4, … }. All they are trees. By a tree we mean a dendrite with finitely many end points, or equivalently, a dendrite containing no points of order ω and having the set of all ramification points finite.

1.3.3 The locally connected fan

Another simple example is the locally connected fan F_{ω}. It is defined as the union of countably many straight line segments of length tending to 0, emanating from a point p and disjoint out of p. Thus F_{ω} is the dendrite with only one ramification point whose order is ω.

The following mapping properties of F_{ω} are known.

1. F_{ω} is universal in the class of all n-ods for n ∈ ℕ.

2. Each open image of F_{ω} is homeomorphic to F_{ω}, [Charatonik et al. 1990, Proposition 9.4, p. 42].

3. A confluent mapping defined on F_{ω} is open if and only if it is light, [Charatonik et al. 1990, Theorem 9.6, p. 42].

1.3.4 The locally connected combs

The next two examples are defined in [Arévalo et al. 2001, p. 3]. Let \overline{pq} stand for the straight line segment from p to q in the plane. In the plane, let a = (0, 0), a_{n} = (1/n, 1/n), b_{n} = (1/n, 0) for each n ∈ ℕ, and c = (−1, 0). Define

W_{R} = \overline{ab_{1}} ∪ ⋃\{\overline{a_{n}b_{n}} : n ∈ ℕ\} and W = \overline{ca} ∪ W_{R}.

The dendrites W_{R} and W are called the comb W_{R} and the comb W , or (generally) locally connected combs .

See Figures A–B.

The dendrites F_{ω}, W_{R} and W are exploited in the following characterizations.

1. A dendrite is a tree if and only if it contains neither a copy of F_{ω} nor of W_{R}, [Arévalo et al. 2001, Theorem 3.1, p. 3].

Figure 1.3.4: ( A ) comb W_{R}

Figure 1.3.4: ( B ) comb W

2. A dendrite has the set of all its end points closed if and only if it contains neither a copy of F_{ω} nor of W, [Arévalo et al. 2001, Corollary 5.4, p. 11].

1.3.5 Universal dendrites

As it was mentioned in Property 1 in 1.3.3 the locally connected fan is universal in the class of all n-ods, where n ∈ ℕ. Now we recall the construction of the Ważewski universal dendrite D_{ω} and some other dendrites that are universal in various classes of these curves.

1.3.6 Ważewski universal dendrite

By the Ważewski universal dendrite D_{ω} we mean a dendrite D_{ω} such that each ramification point of D_{ω} is of order ω and for each arc A ⊂ D_{ω} the set of ramification points of D_{ω} which belong to A is dense in A. Its construction, known from [Whyburn 1942, Chapter K, p. 137] (compare also [Menger 1932, Chapter X, Section 6, p. 318]), is the following.

Let X_{1}^{(ω)} = F_{ω}. At the midpoint c of each maximal free arc contained in X_{1}^{(ω)} (obviously the arc is a straight line segment) attach a sufficiently small copy of F_{ω} so that c is the only common point of X_{1}^{(ω)} and of the attached copy. Denote by X_{2}^{(ω)} the union of X_{1}^{(ω)} and of all attached copies. Thus X_{2}^{(ω)} is a dendrite. At the midpoint of each maximal free arc contained in X_{2}^{(ω)} we perform the same construction, i.e., we attach a sufficiently small copy of F_{ω} so that m is the only common point of X_{2}^{(ω)} and of the attached copy. Denote by X_{3}^{(ω)} the union of X_{2}^{(ω)} and of all attached copies. Thus X_{3}^{(ω)} is a dendrite. Continuing in this way we obtain an increasing sequence of dendrites

Figure 1.3.6: ( A ) Ważewski universal dendrite D_{ω}

X_{1}^{(ω)} ⊂ X_{2}^{(ω)} ⊂ ⋯ ⊂ X_{n}^{(ω)} ⊂ X_{n+1}^{(ω)} ⊂ . . .. The construction can be done in the plane in such a way that the limit continuum D_{ω} defined by

D_{ω} = cl (⋃{X_{n}^{(ω)} : n ∈ ℕ})

is again a dendrite. See Figure A.

For another construction of D_{ω} (using inverse limits) see [Nadler 1992, 10.37, p. 181-185].

The following properties of D_{ω} are known.

1. D_{ω} is universal in the class of all dendrites (see e.g. [Nadler 1992, 10.37, p. 181-185]).

2. D_{ω} is embeddable in the plane (in fact, it is constructed in the plane).

3. Each open image of D_{ω} is homeomorphic to D_{ω} (see [Chaaratonik 1980, Theorem 1, p. 490]).

4. D_{ω} is homogeneous with respect to monotone mappings, [Charatonik 1991, Theorem 7.1, p. 186].

For other mapping properties of D_{ω}, in particular ones related to the action of the group of autohomeomorphisms on D_{ω}, see [Charatonik 1995].

1.3.7 Universal dendrites of order m

Let m ∈ {3, 4, …, ω}. By the standard universal dendrite of order m we mean a dendrite D_{m} such that each ramification point of D_{m} is of order m and for each arc A ⊂ D_{m} the set of ramification points of D_{m} which belong to A is dense in A. Their constructions, known again from [Ważewski 1923, Chapter K, p. 137] (see also [Charatonik 1991, (4), p. 168]; for the inverse limit construction see [Chaaratonik 1980, p. 491]), mimic that of the Ważewski universal dendrite

Figure 1.3.7: ( A ) standard universal dendrite D_{3}

D_{ω}, but instead of copies of F_{ω} we use copies of m-ods at each step of the construction. See Figures A–C for the standard universal dendrites D_{3}, D_{4} and D_{6}.

The standard universal dendrites D_{m} have the following properties.

1. For each m ∈ {3, 4, …, ω} D_{m} is universal in the class of all dendrites for which the order of their ramification points is less than or equal to m, see e.g. [Menger 1932, Chapter 10, §6, p. 322].

2. If m, n ∈ ℕ with 3 ≤ m < n, then there exists an open mapping of D_{n} onto D_{m}, [Chaaratonik 1980, Theorem 2, p. 492].

3. Among all standard universal dendrites D_{m} only D_{3} and D_{ω} are homeomorphic with all their open images, [Chaaratonik 1980, Corollary, p. 493].

4. For each m ∈ {3, 4, …, ω} a monotone surjection of D_{m} onto itself is a near homeomorphism if and only if m = 3, [Charatonik 1991, Corollary 5.5, p. 178].

1.3. DENDRITES 15 Figure 1.3.7: ( B ) standard universal dendrite D_{4} 16 CHAPTER 1. LOCALLY CONNECTED CONTINUA

Figure 1.3.7: ( C ) standard universal dendrite D_{6} 5. Any two standard universal dendrites D_{m} and D_{n} of some orders m, n ∈

{3, 4, …, ω} are monotonely equivalent, [Charatonik 1991, Corollary 6.6, p. 180].

6. For each m ∈ {3, 4, …, ω} the dendrite D_{m} is monotonely homogeneous [Charatonik 1991, Theorem 7.1, p. 186].

Other mapping properties of the standard universal dendrites D_{m} can be found e.g. in [Chaaratonik 1980], [Charatonik 1991], [Charatonik 1995], [Charatonik et al. 1997a], [Charatonik et al. 1998], [Charatonik et al. 1994] and [Charatonik W.J. et al. 1994].

1.3.8 Other universal dendrites

For a given nonempty set S ⊂ {3, 4, …, ω} we denote by D_{S} any dendrite X satisfying the following two conditions:

(a) if p is a ramification point of X, then ord (p, X) ∈ S;

(b) for each arc A contained in X and for every m ∈ S there is in A a point p with ord (p, X) = m.

It is shown in [Charatonik W.J. et al. 1994, Theorem 6.2, p. 229] that the dendrite D_{S} is topologically unique, i.e., if two dendrites satisfy conditions (a) and (b) with the same set S ⊂ {3, 4, …, ω}, then they are homeomorphic. The dendrite D_{S} is called the standard universal dendrite of orders in S . If S is a singleton {m}, then D_{S} is just the standard universal dendrite D_{m} defined previously.

The following properties of dendrites D_{S} are known (see [Charatonik W.J. et al. 1994, Theorems 6.4 and 6.6-6.8, p. 230; Corollary 6.10, p. 232]).

1. For any nonempty subset S ⊂ {3, 4, …, ω}, the dendrite D_{S} is strongly pointwise self-homeomorphic.

2. If ω ∈ S, then the dendrite D_{S} is universal for the family of all dendrites.

3. If the set S is finite with max S = m, then D_{S} is universal for the family of all dendrites having orders of ramification points at most m.

4. If the set S is infinite and ω ∉ S, then D_{S} is universal for the family of all dendrites having finite orders of ramification points.

5. Nonconstant open images of standard universal dendrites D_{S} are homeomorphic to D_{S} if and only if S is a nonempty subset of {3, ω}.

6. For any nonempty subset S ⊂ {3, 4, …, ω} and for an arbitrary dendrite Y there exists a monotone mapping from D_{S} onto Y, [Charatonik et al. 1998, Theorem 2.22, p. 239].

7. For any nonempty subset S ⊂ {3, 4, …, ω} the dendrite D_{S} is monotonely homogeneous, [Charatonik 1996, Theorem 3.3, p. 292].

1.3.9 The dendrite P

In connection with the characterization of dendrites having closed set of their end points (see Property 2 in 1.3.4 above) the classes of those dendrites that do not contain a copy of F_{ω} and those that do not contain a copy of the comb W are of some interest (see [Arévalo et al. 2001, p. 8]). The first one is just the class of dendrites with finite orders of ramification points, and it is known to have a universal element according to Property 4 in 1.3.8. The other class has also a universal element P. Its construction, given in [Arévalo et al. 2001, p. 9] is the following.

Let p be any point in the plane ℝ^{2} and let p_{1}, p_{2}, … be a sequence of points in ℝ^{2} tending to p and such that no three of the points p, p_{1}, p_{2}, … are collinear. Define P_{1} = ⋃\{\overline{pp_{n}} : n ∈ ℕ\}. Then P_{1} is homeomorphic to F_{ω}. For every n, m ∈ ℕ, let p_{nm} be a point in ℝ^{2} \ P_{1} such that for any fixed n the sequence p_{n1}, p_{n2}, … tends to p_{n}, no three of the points p_{n}, p_{n1}, p_{n2}, … are collinear, and that the continuum P_{2} = P_{1} ∪ ⋃\{\overline{p_{n}p_{nm}} : n, m ∈ ℕ\} is a dendrite. We continue constructing dendrites P_{3}, P_{4}, … in the same manner, such that we get an increasing sequence of dendrites P_{1} ⊂ P_{2} ⊂ P_{3} ⊂ ⋯ ⊂ P_{n} ⊂ P_{n+1} ⊂ … in ℝ^{2} having the property that the closure of their union is also a dendrite. Finally define

P = cl (⋃{P_{n} : n ∈ ℕ}).

The following properties of the dendrite P are proved in [Arévalo et al. 2001, Theorems 4.5 and 4.6, p. 10].

1. A dendrite X is homeomorphic to the dendrite P if and only if each of the following conditions is satisfied:

(a) X contains no copy of W;

(b) the set of all end points of X is contained in the closure of the set of all ramification points of X;

(c) all ramification points of X are of order ω.

2. The dendrite P is universal in the class of all dendrites containing no copy of W.

See Figure A.

1.3.10 Self-homeomorphic dendrites

Below we present some examples of dendrites that illustrate relations between variants of self-homeomorphic properties.

1. There is a strongly pointwise self-homeomorphic dendrite which is not of the form D_{S} for some S ⊂ {3, 4, …, ω}, see [Charatonik W.J. et al. 1994, Example 6.11, p. 232, and Fig. 4, p. 233]. Note that, according to Property 1 in 1.3.8 each dendrite D_{S} is

Figure 1.3.9: ( A ) dendrite P

strongly pointwise self-homeomorphic. This example shows that the opposite implication does not hold. Some additional properties of the example are shown in [Pyrih 1999b, Example 2.1, p. 152]. See Figure A.

2. There is a strongly self-homeomorphic and pointwise self-homeomorphic dendrite which is not strongly pointwise self-homeomorphic, see [Charatonik W.J. et al. 1994, Example 6.12, p. 233].

3. There is a strongly self-homeomorphic dendrite which is not pointwise self-homeomorphic, see [Pyrih 1999a, Example 2.1, p. 572]. Dendrites having this property are studied in [Charatonik et al. XXXXc].

See Figure B.

1.3.11 Monotone equivalence and monotone homogeneity

As it was mentioned previously, any two universal dendrites D_{m} and D_{n} are monotonely equivalent (see Property 5 in 1.3.7). However, the family of all dendrites which are monotonely equivalent to a dendrite D_{m} contains also other members. To characterize the family we have to construct a special dendrite, denoted L_{0}.

1.3.12 Omiljanowski dendrite

The Omiljanowski dendrite has been constructed by K. Omiljanowski in [Charatonik 1991, Example 6.9, p. 182] (see also [Charatonik 1991, Remark 6.11 and Theorem 6.12, p. 183]). It is defined as the closure of the union of an increasing sequence of dendrites in the plane. We start with the unit straight 20 CHAPTER 1. LOCALLY CONNECTED CONTINUA

Figure 1.3.10: ( A ) an example of self-homeomorphic dendrite 1.3. DENDRITES 21

Figure 1.3.10: ( B ) an example of strongly self-homeomorphic dendrite

Figure 1.3.12: ( A ) dendrite L_{0}

line segment denoted by L_{1}. Divide L_{1} into three equal subsegments and in the middle of them, M, locate a thrice diminished copy of the Cantor ternary set C. At the midpoint of each contiguous interval K to C (i.e., a component K of M \ C) we erect perpendicularly to L_{1} a straight line segment whose length equals length of K. Denote by L_{2} the union of L_{1} and of all erected segments (there are countably many of them, and their lengths tend to zero). We perform the same construction on each of the added segments: divide such a segment into three equal parts, locate in the middle part M a copy of the Cantor set C properly diminished, at the midpoint of any component K of M \ C construct a perpendicular to K segment as long as K is, and denote by L_{3} the union of L_{2} and of all attached segments. Continuing in this manner we get a sequence of dendrites L_{1} ⊂ L_{2} ⊂ L_{3} ⊂ . . .. Finally we put

L_{0} = cl (⋃{L_{i} : i ∈ ℕ})

and call it a dendrite L_{0} . See Figure A.

The Omiljanowski dendrite L_{0} has the following properties, [Charatonik 1991, Example 6.9, p. 182].

1. All ramification points of L_{0} are of order 3.

2. The set R(L_{0}) of all ramification points of L_{0} is discrete (thus nowhere dense).

3. The set of all end points of L_{0} is nowhere dense.

4. For each maximal arc A in L_{0} the closure of the set A ∩ R(L_{0}) contains a homeomorphic copy of the Cantor set.

5. L_{0} is monotonely equivalent to the dendrite D_{3}.

6. The following conditions are equivalent for a dendrite X (see [Charatonik 1991, Theorem 6.14, p. 185] and [Charatonik et al. 1994, Theorem 5.35, p. 17]; compare [Charatonik et al. 1998, Theorem 2.20, p. 238]):

(a) X is monotonely equivalent to D_{3}; (b) X is monotonely equivalent to D_{ω};

(c) X is monotonely equivalent to D_{m} for each m ∈ {3, 4, …, ω};

(d) X is monotonely equivalent to D_{S} for each nonempty set S ⊂ {3, 4, …, ω};

(e) X is monotonely equivalent to every dendrite Y having dense set R(Y) of its ramification points;

(f) X is monotonely equivalent to some dendrite Y having dense set R(Y) of its ramification points;

(g) X contains a homeomorphic copy of every dendrite L such that its set R(L) of ramification points is discrete and consists of points of order 3 exclusively;

(h) X contains a homeomorphic copy of the dendrite L_{0}.

According to Property 7 in 1.3.8 (see also Property 6 in 1.3.7) any dendrite D_{S} is monotonely homogeneous. Another, less restrictive, sufficient condition for monotone homogeneity of a dendrite is the following (see [Charatonik et al. 1997a, Proposition 15, p. 364]).

7. If for a dendrite X the set R(X) of its ramification points is dense in X, then X is monotonely homogeneous.

The condition cl R(X) = X, being sufficient, is far from being necessary. Namely the Omiljanowski dendrite L_{0} has the set R(L_{0}) discrete, and it is monotonely homogeneous. Moreover, the following statement holds, [Charatonik et al. 1997a, Proposition 20, p. 366].

8. If a dendrite X contains a homeomorphic copy of L_{0}, then X is monotonely homogeneous.

It would be interesting to know if the converse to the above statement holds true, i.e., if containing the dendrite L_{0} characterizes monotonely homogeneous dendrites. In other words, we have the following question.

Question. Does every monotonely homogeneous dendrite contain a homeomorphic copy of L_{0}?

The above question is closely related to a more general problem. Problem. Give any structural characterization of monotonely homoge-

neous.

1.3.13 Chaotic and rigid dendrites

We start presentation of examples related to chaoticity with an example of a dendrite which, while not chaotic, is closely related to the property, and which has been considered as a pattern to construct other examples having various related properties.

1.3.14 Miller dendrite

Recall that a space is said to be incompressible if it is homeomorphic to no proper subspace of itself. The circle S^{1} is an example of such a space. In 1925 Zarankiewicz [Zarankiewicz 1925] asked whether there exists an incompressible dendrite. The question has been answered in the positive in 1932 by E. W. Miller [Miller 1932]. The reader is referred to the original Miller’s paper for the (rather long) construction of the example. The main idea of the proof is based on the following implication, [Miller 1932, Theorem, p. 831].

If a dendrite X contains a subset K such that

A. each point of K is fixed with respect to any homeomorphism of X onto a subcontinuum of X;

B. each point of X which is not an end point of X lies in an arc contained in X and having its end points in K,

then X is incompressible.

The Miller dendrite S has the following properties, [Miller 1932].

1. S contains a subset K having properties A and B above, so it is incompressible.

2. S is not chaotic.

3. S contains open arcs as open subsets, hence it is not rigid.

1.3.15 Dendrites of de Groot-Wille type

An example of a rigid dendrite has been roughly described in [J. de Groot et al. 1958]. Its construction is recalled below (see also [Charatonik 1979, Chapter III, Section 5 , p. 227]).

For any i ∈ ℕ let T_{i} denote the i-od being the union of i straight line unit segments emanating from a point called the origin of T_{i}. We proceed by induction. Define Y_{0} as the unit segment. Let x be the midpoint of Y_{0}, and define Y_{1} as the union of Y_{0} and a diminished copy of T_{j_{1}} = T_{1} so that the diameter of this copy is less than 1/2 and that Y_{0} ∩ T_{1} = {x}. Assume that a tree Y_{n} is defined such that it is the union of finitely many straight line segments and contains (properly diminished) copies of T_{1}, T_{2}, …, T_{j_{n}}, i.e., of the first j_{n} terms of the sequence {T_{i}}. To define Y_{n+1} consider all maximal free segments in Y_{n}. Note that there are finitely many, say m(n) of them. Let x denote the midpoint of any of these segments. With each point x so defined we associate, in a one-to-one way, a set T_{i} taken from the m(n) consecutive terms of the sequence {T_{i}}, i.e., we use in this step of the construction the sets T_{j_{n}+1}, T_{j_{n}+2}, …, T_{j_{n+1}}, where j_{n+1} = j_{n} + m(n). We take each midpoint as the origin of a properly diminished copy of T_{i}, where i ∈ {j_{n} + 1, j_{n} + 2, …, j_{n+1}} in such a way that the diameter of the copy of T_{i} is less than 1/2^{n+1} and that Y_{n} has only the

Figure 1.3.15: ( A ) Y_{3}

point x in common with the attached copy of T_{i}. All this can clearly be done so carefully that the resulting set Y_{n+1} is a tree and the limit continuum

Y = cl (⋃{Y_{n} : n ∈ ℕ})

is a dendrite. See Figure A.

The following properties of the dendrite Y are shown in [Charatonik 1979, Statement 10, p. 229].

1. The set of all end points of Y is dense in Y.

2. For each n ∈ ℕ the set of all points of Y that are of order at least n is dense in Y.

3. For each natural n ≥ 3 the dendrite Y contains exactly one point of order n.

4. For each point p ∈ Y and for each component C of Y \ {p} the continuum C ∪ {p} contains a homeomorphic copy of Y.

5. Y is not strongly rigid.

6. Y is chaotic.

Recall that for any dendrite conditions 1 and 3 imply 5 and 6, see [Charatonik 1999, Theorem 13, p. 22]. However, neither the above mentioned conditions, nor a rough description given in [J. de Groot et al. 1958], nor the one presented in [Charatonik 1979] lead to a uniquely determined dendrite, because the constructed dendrite Y depends on a function that assigns the consecutive i-ods T_{i} (used in the succesive steps of the construction) to the midpoints of the maximal free arcs in the trees Y_{n}. Thus we refer to any of the dendrites Y obtained in this way as to a dendrite of de Groot-Wille type .

An example of a dendrite of de Groot-Wille type that is chaotic but not openly chaotic is constructed in [Charatonik 2000, Theorem 3.10, p. 646].

1.3.16 Modified Miller dendrites

The methods used by Miller to construct his example S has also been applied to get a modification D of S in [Charatonik 1979, Chapter IV, p. 230] (an outline of this construction is in [Caratonik 1977]). Below we summarize properties of this example, and recall an extension of this construction.

1. There exists a dendrite D such that:

(a) each point of D is of order nor greater than 4;

(b) for each n ∈ {1, 2, 3, 4} the set of all points of D which are of order n is dense in D;

(c) D is strongly rigid;

(d) D is chaotic.

2. For any two integers m and n with 3 ≤ m < n there exists a dendrite X(m, n) such that (see [Charatonik et al. 1996, Theorem 5.5, p. 185] and compare also [Charatonik 1999, Theorem 27, p. 24]):

(a) ord (x, X(m, n)) ∈ {1, 2, m, n} for each x ∈ X(m, n).

(b) Every arc in X(m, n) contains a point of order m in X(m, n).

(c) Each of the sets: of all end points of X(m, n), of all points of order m in X(m, n) and of all points of order n in X(m, n) is dense in X(m, n).

(d) X(m, n) is strongly chaotic.

Dendrite D is called a modified Miller dendrite . (For a further modification of the above construction that leads to chaotic, strongly rigid, openly rigid and not strongly chaotic dendrites, see [Charatonik 2000, Theorem 3.14, p. 650].)

Finally recall dendrites constructed in [Charatonik 1999, Examples 33 and 35, p. 28].

3. For each natural n ≥ 5, there exists a strongly rigid and not chaotic dendrite, all points of which are of order at most n ([Charatonik 1999, Example 33, p. 28]).

4. There exists a rigid dendrite which is neither chaotic nor strongly rigid ([Charatonik 1999, Example 35, p. 28]).

1.3.17 Dendrites with the closed set of end points

Dendrites in this class have been already characterized (see Property 2 in 1.3.4). Besides any tree, the locally connected comb W_{R} comb W_{R} is an example of a dendrite having the set of its end points closed (and countable). Below we recall a member of the considered class having uncountably many end points.

Figure 1.3.18: ( A ) Gehman dendrite

1.3.18 Gehman dendrite

One of the classical examples of dendrites with its set of end points closed is the Gehman dendrite . It can be described as a dendrite G having the set E(G) homeomorphic to the Cantor ternary set C in [0, 1] such that all ramification points of G are of order 3 (see [Gehman 1925, the example on p. 42]; see also [Nikiel 1983, p. 422-423] for a geometrical description; compare [Nikiel 1989, p. 82] and [Nadler 1992, Example 10.39, p. 186]). See Figure A.

The Gehman dendrite has the following properties.

1. The set R(G) of all ramification points of G is discrete.

2. E(G) = cl (R(G)) \ R(G).

3. Each dendrite with an uncountable set of its end points contains a homeomorphic copy of the Gehman dendrite, [Arévalo et al. 2001, Proposition 6.8, p. 16].

4. If a continuum contains the Gehman dendrite, then it does not have the periodic-recurrent property, [Charatonik 1998, Theorem 3.3, p. 136].

5. A dendrite X contains the Gehman dendrite if and only if X does not have the periodic-recurrent property, [Illanes 1998, Theorem 2, p. 222].

1.3.19 Modifications of the Gehman dendrite

The concept of the Gehman dendrite has been generalized in [Arévalo et al. 2001, Section 4, p. 4-10] as follows.

1. For each natural number n ≥ 3 there exists topologically unique dendrite G_{n} dendrite G_{n} such that

(a) ord (p, G_{n}) = n for each point p ∈ R(G_{n});

Figure 1.3.19: ( A ) G_{4}

(b) E(G_{n}) is homeomorphic to the Cantor ternary set.

(Note that G_{3} is just the Gehman dendrite G.)

2. There exists a dendrite G_{ω} such that

(a) ord (p, G_{ω}) is finite for each point p ∈ G_{ω};

(b) E(G_{ω}) is homeomorphic to the Cantor ternary set;

(c) for each natural number n and for each maximal arc A contained in G_{ω} there is a point q ∈ A such that ord (q, G_{ω}) ≥ n.

See Figure A.

The dendrites G_{n} and G_{ω} have the following universality properties.

3. For each natural number n ≥ 3 the dendrite G_{n} is universal for the class of all dendrites X such that cl E(X) = E(X) and that ord (p, X) ≤ n for each point p ∈ X.

4. Each dendrite G_{ω} is universal for the class of all dendrites with the closed set of end points.

1.3.20 Mapping hierarchy of dendrites

A classification of topological spaces from the standpoint of the theory of mappings (called also a mapping hierarchy of spaces) is defined in [Borsuk 1959], and it is investigated for dendrites in [Charatonik et al. 1994]. Some questions asked there have already been answered. We recall some examples related to this subject.

1. There exist two dendrites X and Y such that {X, Y } has no infimum and no supremum with respect to the class of monotone mappings (see [Charatonik et al. 1994, Example 5.45, p. 18]).

1.4. CYCLIC EXAMPLES OF LOCALLY CONNECTED CONTINUA 29

2. There exist dendrites X and Y, both admitting open mappings onto arcs, such that Y ⊂ X and there is no open mapping from X onto Y (see [Charatonik et al. 1994, Example 6.67, p. 34])

Indeed, in an arm A of a simple triod Y fix a sequence {p_{n}} of points of A converging to the center of the triod, and take a sequence {A_{n}} of arcs such that Y ∩A_{n} = {p_{n}} and that lim diam_{n} = 0. Then X = Y ∪⋃{A_{n} : n ∈ ℕ} is the needed dendrite.

3. There exists a dendrite which does not admit any open mapping onto an arc and which is the union of two subdendrites admitting open mappings onto an arc (see [Charatonik et al. 1994, Example 6.69, p. 34]).

4. There is a bounded chain of dendrites (ordered with respect to open mappings) that has no supremum (see [Podbrdský et al. 2001]).

5. There is in the plane an uncountable family of dendrites every two members of which are incomparable by r-mappings (see [Sieklucki 1959]).

6. There is a family of dendrites ordered with respect to r-mappings similarly to the segment (see [Sieklucki 1961]).

1.4 Cyclic examples of locally connected continua

(a) Claytor examples of nonplanable curves

(b) Menger compacta, Sierpiński universal plane curve, Menger universal curve; homogeneity of these continua

(c) Other examples

1.4.1 Menger universal continua

The Menger universal continuum M_{n}^{m}, for 1 ≤ n < m [Menger 1926], can be defined as follows [Engelking 1978, pp. 121–122].

Let F_{0} = {I^{m}}. Inductively, suppose a collection F_{k} of cubes has been defined for k ≥ 0. Divide each cube D ∈ F_{k} into 3^{m(k+1)} congruent cubes with edges of length \frac{1}{3^{m(k+1)}} . If F_{k+1}(D) is the collection of all these cubes that intersect n-dimensional faces of D, then let F_{k+1} = ⋃{ F_{k+1}(D) : D ∈ F_{k} }.

Define

M_{n}^{m} = ⋂_{k=0}^{∞} (⋃ F_{k}) .

The most popular continua are M_{1}^{2}, widely known as the Sierpiński universal plane curve or the Sierpiński carpet [Sierpiński 1916], and M_{1}^{3}, called the Menger universal curve [Menger 1926]. These particular curves are discussed separately.

Continuum M_{n}^{2n+1} can also be obtained as the inverse limit \underleftarrow{lim}(X_{i}, p_{i}^{i+1}) in the following way. Let {G_{1}, G_{2}, … } be an open base of S^{n} consisting of open n-cells such that diam_{i} → 0. Put X_{1} = S^{n} and X_{i+1} = (X_{i} × {0, 1})/ ∼, where relation ∼ identifies points (x, 0), (x, 1) iff p_{1}^{i}(x) ∉ G_{i}. The bonding maps p_{i}^{i+1} are natural retractions [Pasynkov 1965], [Bestvina 1988, pp. 98–99].

1. M_{n}^{m} is the universal space in the class of all compacta of dimension ≤ n which embed in ℝ^{m} [Štanko 1971] ([Menger 1926] for M_{m−1}^{m}). M_{m−1}^{m} is universal in the class of all (m − 1)-dimensional subsets of ℝ^{m} ([Engelking 1978, Problem 1.11.D]), and M_{n}^{2n+1} in the class of all n-dimensional separable metric spaces [Bothe 1963].

2. A continuum X is homeomorphic to M_{m}^{m+1} if and only if X can be embedded in the (m+1)-sphere S^{m+1} in such a way that S^{m+1} \X has infinitely many components C_{1}, C_{2}, … such that diam_{i} → 0, bd C_{i} ∩ bd C_{j} = ∅ for i ≠ j, bd C_{i} is an m-cell for each i and ⋃_{i=1}^{∞} bd C_{i} is dense in X (see [Cannon 1973] and [Chigogidze et al. 1995, Theorem 6.1.2, p. 74]).

3. A continuum X is homeomorphic to M_{n}^{2n+1} if and only if X is n-dimensional, C^{n−1} and LC^{n−1}-space with the disjoint n-disks property (DD^{n}P property) [Bestvina 1988, Corollary 5.2.3, p.98].

Equivalently, X is homeomorphic to M_{n}^{2n+1} if and only if X is n-dimensional, C^{n−1} and LC^{n−1}-space such that any continuous map from a compact space of dimension ≤ n into M_{n}^{2n+1} is a uniform limit of embeddings (Z-embeddings) [Bestvina 1988, Theorem 2.3.8, p. 36].

4. (a) If Z ⊂ M_{n}^{2n+1} is a Z-setZ-set, then, for every open neighborhood U ⊂ M_{n}^{2n+1} of Z and every ϵ > 0, there is δ > 0 such that if Z′ ⊂ M_{n}^{2n+1} is a Z-set and h : Z → Z′ is a δ-homeomorphism, then h extends to an ϵ-homeomorphism h^{∗} : M_{n}^{2n+1} → M_{n}^{2n+1} such that h^{∗}|M_{n}^{2n+1} \ U = identity [Bestvina 1988, Theorem 3.1.1, p. 65].

(b) For every ϵ > 0, there exists δ > 0 such that if Z, Z′ are Z-setsZ-set in M_{n}^{2n+1} and h : Z → Z′ is a δ-homeomorphism, then there is an ϵhomeomorphism h^{∗} : M_{n}^{2n+1} → M_{n}^{2n+1} extending h [Bestvina 1988, Theorem 3.1.3, p. 71].

(c) Every homeomorphism between Z-sets in M_{n}^{2n+1} extends to an autohomeomorphism of M_{n}^{2n+1} [Bestvina 1988, Corollary 3.1.5, p. 72].

It follows that M_{n}^{2n+1} is strongly locally homogeneous and countably dense homogeneous.

5. If m < 2n+1, then M_{n}^{m} is not homogeneous [Lewis 1987] (for m ≥ 2n+1 the homogeneity of M_{n}^{m} follows from the property above).

6. M_{n}^{2n+1} × X is not 2-homogeneous for an arbitrary continuum X [Kuperberg et al. 1995].

7. The groups of autohomeomorphisms of M_{n}^{2n+1} and M_{n}^{n+1} are Polish and one-dimensional (see [Oversteegen et al. 1994, Corollary 6] and Property 2).

The group is totally disconnected for M_{n}^{2n+1} (see [Brechner 1966, Theorem 1.3] for n = 1 and its natural extension for any n).

8. The Polish group of autohomeomorphisms of M_{n}^{m} is at most one-dimensional [Oversteegen et al. 1994].

9. The group of autohomeomorphisms of M_{n}^{2n+1} is simple (see [Chigogidze et al. 1995, Theorem 3.2.4]).

10. Every autohomeomorphism of M_{n}^{2n+1} is a composition of two homeomorphisms, each of which is the identity on some nonempty open set [Sakai 1994].

11. For each t ∈ [n, 2n + 1], there is a copy of M_{n}^{2n+1} ⊂ ℝ^{2n+1} with the Hausdorff dimension t [Chigogidze et al. 1995, Theorem 4.2.6].

12. Every Z-set in M_{n}^{2n+1} is the fixed point set of some autohomeomorphisms of M_{n}^{2n+1} [Sakai 1997].

13. Any C^{n−1} and LC^{n−1}-continuum is the image of M_{n}^{2n+1} under a UV^{n−1}- map [Bestvina 1988, Theorem 5.1.8, p. 95].

14. If a Polish space X contains a topological copy M of M_{n}^{2n+1} or M_{n}^{n+1}, then the space of all copies of M in X with the Hausdorff metric is not a G_{δσ}-set [Krupski XXXXb].

1.4.2 Sierpiński carpet

The Sierpiński universal plane curve or the Sierpiński carpet [Sierpiński 1916] M_{1}^{2} is a well known fractal obtained as the set remaining when one begins with the unit square I^{2} and applies the operation of dividing it into 9 congruent squares and deleting the interior of the central one, then repeats this operation on each of the surviving 8 squares, and so on. See Figure A.

1. M_{1}^{2} is universal in the class of all at most one-dimensional subsets of the plane (equivalently, of all boundary subsets of the plane) [Sierpiński 1916], [Sierpiński 1922].

2. The following statements are equivalent:

(a) X is homeomorphic to M_{1}^{2};

(b) X is a locally connected plane curve that contains no local cut points;

(c) X is a continuum embeddable in the plane in such a way that ℝ^{2} \ X has infinitely many components C_{1}, C_{2}, … such that diam_{i} → 0, bd C_{i} ∩ bd C_{j} = ∅ for i ≠ j, bd C_{i} is a simple closed curve for each i and ⋃_{i=1}^{∞} bd C_{i} is dense in X [Whyburn 1958].

32 CHAPTER 1. LOCALLY CONNECTED CONTINUA Figure 1.4.2: ( A ) Sierpiński carpet 3. A complete metric space X contains a topological copy of M_{1}^{2} if and only

if X contains a subset with the bypass property [Prajs 1998a].

4. The group of all autohomeomorphisms of M_{1}^{2} has exactly two orbits: one of them is the union of all simple closed curves which are the boundaries of complementary domains of M_{1}^{2} [Krasinkiewicz 1969].

The group is a Polish topological group which is totally disconnected and one-dimensional (see [Brechner 1966, Theorem 1.2] and Property 7 in 1.4.1).

5. Any homeomorphism between Cartesian products of copies of the Sierpiński carpet is factor preserving [Kennedy Phelps 1980]. Consequently, no such product is homogeneous.

6. The Sierpiński carpet can be continuously decomposed into pseudo-arcs such that the decomposition space is homeomorphic to the carpet [Prajs 1998b, Corollary 18], [Seaquist 1995]. In fact, M_{1}^{2} is the only planar locally connected curve admitting such a decomposition [Prajs 1998b, Corollary 18].

7. The Sierpiński carpet is homogeneous with respect to monotone open mappings [Prajs 1998b, Corollary 24], [Seaquist 1999]. Moreover, every continuum which is locally homeomorphic to M_{1}^{2} (i. e., M_{1}^{2}-manifold) is homogeneous with respect to monotone open mappings [Prajs 1998b, Theorem 23].

8. M_{1}^{2} is homogeneous with respect to the class of simple mappings [Charatonik 1984].

9. If C is a curve, then the set of all mappings f : C → ℝ^{2} such that f(C) is homeomorphic to M_{1}^{2} is a residual subset of the space (ℝ^{2})^{C} of all mappings of C into ℝ^{2} with the uniform convergence metric.

If C(X) is the hyperspace of all subcontinua of a compact space X and C_{1}(X) its subspace of all curves, then the set

{ (C, f) ∈ C(X) × (ℝ^{2})^{X} : f(C) is homeomorphic to M_{1}^{2} }

is residual in C(X)×(ℝ^{2})^{X}; in other words, almost all mappings in (ℝ^{2})^{X} map almost all curves in X onto copies of M_{1}^{2}, where ”almost all” means all with except of a subset of the first category in corresponding spaces [Mazurkiewicz 1938].

10. If a locally compact space X contains a topological copy of the Sierpiński carpet, then the space of all copies of the Sierpiński carpets in X with the Hausdorff metricHausdorff metric is a true absolute F_{σδ}-set [Krupski XXXXb].

1.4.3 Menger universal curve

The Menger universal curve M_{1}^{3} is the subset of the unit cube I^{3} whose projections onto faces of the cube are the Sierpiński carpets, i.e.,

M_{1}^{3} = { (x, y, z) ∈ I^{3} : (x, y) ∈ M_{1}^{2}, (y, z) ∈ M_{1}^{2}, (x, z) ∈ M_{1}^{2} }

[Menger 1926]. Other descriptions can be found in 1.4.1 (see also [Mayer et al. 1986, pp. 5–6, 8–9]). See Figure A.

1. The following statements are equivalent:

(a) X is homeomorphic to M_{1}^{3};

(b) X is a locally connected curve with no local cut points and no planar open nonempty subsets [Anderson 1958, Theorem XII, p. 13];

(c) X is a homogeneous locally connected curve different from a simple close curve [Anderson 1958, Theorem XIII, p. 14];

(d) X is a locally connected curve with the disjoint arcs property (see Property 3 in 1.4.1);

(e) X is a locally connected curve and each arc in X is approximately non-locally-separating arc and has empty interior in X [Krupski et al. XXXXa, Theorem 3, p. 86].

2. M_{1}^{3} is universal in the class of all metric separable spaces of dimension ≤ 1 [Menger 1926].

3. Z-setsZ-set in M_{1}^{3} coincide with non-locally-separating closed subsets of M_{1}^{3}.

If Z is a separable metric space of dimension ≤ 1, then Z can be embedded as a non-locally-separating subset of M_{1}^{3} [Mayer et al. 1986, Theorem 6.1, p. 42].

4. If Z is a closed subset of a metric space X, dim(X\Z) ≤ 1 and f : Z → M_{1}^{3} is a continuous mapping with non-locally separating image f(Z), then f can be extended to a map g : X → M_{1}^{3} such that g|X \ Z is an embedding into M_{1}^{3} \ f(Z) [Mayer et al. 1986, Theorem 6.4, p. 44].

5. Every continuous surjection h between non-locally separating closed subsets Z, Z′ of M_{1}^{3} extends to a mapping h^{∗} of M_{1}^{3} such that h^{∗}|M_{1}^{3} \ Z is a homeomorphism onto M_{1}^{3} \ Z′ [Mayer et al. 1986, Corollary 6.5, p. 44]; moreover, for every ϵ > 0 there exists δ > 0 such that if h is a δ-homeomorphism, then h^{∗} can be taken as an ϵ-homeomorphism (see Property 4b in 1.4.1).

6. For each locally connected continuum X, there exist open surjections f : M_{1}^{3} → X and g : M_{1}^{3} → X such that f^{−1}(x) is homeomorphic to M_{1}^{3} and g^{−1}(x) is a Cantor set, for any x ∈ X [Wilson 1972].

1.4. CYCLIC EXAMPLES OF LOCALLY CONNECTED CONTINUA 35 Figure 1.4.3: ( A ) Menger universal curve 36 CHAPTER 1. LOCALLY CONNECTED CONTINUA

7. Any compact 0-dimensional group G acts freely on M_{1}^{3} so that the orbit space M_{1}^{3}/G is homeomorphic to M_{1}^{3} [Anderson 1957, Theorem 1].

8. If a locally compact space X contains a topological copy of M_{1}^{3}, then the space of all copies of M_{1}^{3} in X with the Hausdorff metricHausdorff metric is a true absolute F_{σδ}-set [Krupski XXXXb].

1.5 2-dimensional locally connected continua

(a) Surfaces, polyhedra

(b) Pontryagin surfaces, Pontryagin disk (sphere)

(c) 2-dimensional AR containing no disk

(d) Borsuk’s hat

(e) Alexander horned sphere

(f) Bing dogbone space

(g) Other examples

1.6 Higher dimensional locally connected con-

tinua

(a) n-dimensional manifolds

(b) Hilbert cube manifolds

(c) Other important examples

1.7 Variations of local connectedness

(a) Hereditary local connectedness

(b) Connectedness im kleinen

(c) Semi-local connectedness

JOINING PROPERTIES

2.1 Cut points and separating points (also lo-

cal) 2.2 Arcwise connectedness; arc components 2.3 Continuum chainability 2.4 Various kinds of connectedness

(λ-connectedness, δ-connectedness, etc.)

38 CHAPTER 2. JOINING PROPERTIES

DECOMPOSABLE AND INDECOMPOSABLE CONTINUA

3.1 Decomposable continua (also hereditary)

3.2 Indecomposable continua (also hereditary)

(a) Lakes of Wada

(b) Knaster type continua; Brouwer-Janiszewski-Knaster (buckethandle) continuum

(c) Other simple examples

(d) The pseudo-arc

3.2.1 Knaster continua

By a Knaster continuum is meant a continuum homeomorphic to the inverse limit \underleftarrow{lim}(I_{n}, f_{n}) of a sequence of unit intervals I_{n} = I with open, non-homeomorphic bonding maps f_{n}. Denote by K the class of all Knaster continua.

The most popular examples of Knaster continua is the buckethandle (sometimes called by dynamists the horseshoe ) B_{1}, where f_{n}(t) = 1 − |2t − 1|, and the double buckethandle B_{2} with

f_{n}(t) = \begin{cases}3t & if 0 ≤ t ≤ \frac{1}{3}, \\ 2 − 3t & if \frac{1}{3} ≤ t ≤ \frac{2}{3}, \\ 3t − 2 & if \frac{2}{3} ≤ t ≤ 1. \end{cases}

40CHAPTER 3. DECOMPOSABLE AND INDECOMPOSABLE CONTINUA

Figure 3.2.1: ( A ) buckethandle

Nice geometric models of B_{1} and B_{2} can be described in the following way (see [Kuratowski 1968, pp. 204–205]). If C denotes the standard Cantor ternary set in I, then B_{1} is homeomorphic to the union of all semi-circles

\{ (x, y) ∈ ℝ^{2} : (x − \frac{1}{2})^{2} + y^{2} = r^{2}, y ≥ 0 \},

where r + \frac{1}{2} ∈ C, and all semi-circles

\{ (x, y) : (x − \frac{5}{2 · 3^{n}})^{2} + y^{2} = r^{2}, y ≤ 0 \},

for n ∈ ℕ, where r + \frac{5}{2·3^{n}} ∈ C ∩ [\frac{2}{3^{n}} , \frac{1}{3^{n−1}} ].

See Figure A.

If E is the quintary Cantor set in I (real numbers in I which can be written in the enumeration system at base 5 without digits 1 and 3), then B_{2} is homeomorphic to the union of all semi-circles

\{ (x, y) : (x − \frac{7}{10 · 5^{n}})^{2} + y^{2} = r^{2}, y ≤ 0 \},

for n ∈ ℕ and r + \frac{7}{10·5^{n}} ∈ E ∩ [\frac{2}{5^{n+1}} , \frac{1}{5^{n}}] and all semi-circles

\{ (x, y) : (x − (1 − \frac{7}{10 · 5^{n}}))^{2} + y^{2} = r^{2}, y ≥ 0 \},

for n ≥ 0 and \frac{2}{5^{n+1}} ≤ \frac{7}{10·5^{n}} − r ≤ \frac{1}{5^{n}}.

See Figure B.

Figure 3.2.1: ( B ) double buckethandle

1. If w_{p} : I → I is the map such that w_{p}(i/p) = 0 for even i, w_{k}(i/p) = 1 for odd i, where i is an integer between 0 and p ∈ ℕ, and w_{p} is linear on each interval [i/p, (i + 1)/p], then K is equal to the family of all continua homeomorphic to inverse limits \underleftarrow{lim}(I_{n}, f_{n}), where, for each n, I_{n} = I and f_{n} = w_{p_{n}} for some p_{n} [Rogers 1970]. So, each K ∈ K is determined by a sequence p = (p_{1}, p_{2}, …) such that K = \underleftarrow{lim}(I_{n}, w_{p_{n}}). We will denote such K by K(p) and call it a p-adic Knaster continuum . Without loss of generality, one can assume that all p_{n}’s are prime.

2. Each p-adic Knaster continuum K is homeomorphic to the quotient of the p-adic solenoid Σ(p) under the relation

a ∼ b ⇔ a = b or a = b^{−1},

where a, b ∈ Σ(p) and b^{−1} is the group inverse of b. A point of K is an end point of K if and only if it is an image under the quotient map of the neutral element or the element of order 2 (if such exists) of the corresponding solenoid (see, e.g., [Krupski 1984b, p. 43]).

3. The class K of all Knaster continua is equal to the class of all arc-like continua with one or two end points having the property of Kelley and arcs as proper non-degenerate subcontinua and which themselves are not arcs [Krupski 1984b]. In particular, all of them are indecomposable.

4. There are 2^{ℵ_{0}} mutually non-homeomorphic members of K [Wa]; in fact, there is a subfamily of K of cardinality 2^{ℵ_{0}} no member of which is an open image of another one [Dȩbski 1985].

5. Any open map between two Knaster continua \underleftarrow{lim}(I_{n}, f_{n}), \underleftarrow{lim}(I_{n}, g_{n}) is the uniform limit of open maps which are induced by maps (I_{n}, f_{n}) → (I_{n}, g_{n}) between the inverse sequences [Eberhart et al. 1999, Theorem 4.8, p. 145].

6. If K ∈ K, then every indecomposable continuum can be mapped onto K [Rogers 1970, Corollary, p. 455].

Moreover, if X is an indecomposable (Hausdorff) continuum, a, b ∈ X and a ≠ b (a and b belong to different composants of X), then there is a continuous surjection f : X → B_{1} such that f(a) ≠ f(b) (f(a) and f(b) belong to different composants of B_{1}, resp.). It follows that any indecomposable continuum embeds in the Cartesian product of copies of B_{1} [Bellamy 1973, Corollaries 1 and 2, p. 305].

7. If K_{1}, K_{2} ∈ K then there are 2^{ℵ_{0}} distinct homotopy types of maps of K_{1} onto K_{2} that map an end point of K_{1} onto an end point of K_{2} [Minc 1999].

8. If K ∈ K and f is a monotone map of K, then f(K) is homeomorphic to K [Krupski 1984b].

9. There is no exactly 2-to-1 map defined on a Knaster continuum [Dȩbski 1992].

10. Knaster continua are absolute retracts for the class of all hereditarily unicoherent continua, i.e., if K ∈ K, X is a hereditarily unicoherent continuum and K ⊂ X, then K is a retract of X. Equivalently, if A is a closed subset of a hereditarily unicoherent continuum X, then every map f from A onto a Knaster continuum can be extended over X ([Maćkowiak 1984] for K = B_{1} and X a Hausdorff hereditarily unicoherent continuum and [Charatonik et al. XXXXa] for arbitrary K ∈ K).

11. Any two points of a plane continuum X can be joined by a hereditarily decomposable subcontinuum if and only if X cannot be mapped onto B_{1} [Hagopian 1974, Theorem 2, p. 133].

12. Every autohomeomorphism of the buckethandle B_{1} is isotopic to some iterate of the shift homeomorphism f : B_{1} → B_{1}, f(x_{1}, x_{2}, …) = (x_{2}, x_{3}, …) [Aarts et al. 1991, Theorem 4.4, p. 204].

Moreover, any autohomeomorphism f of a Knaster continuum \underleftarrow{lim}(I_{n}, w_{p_{n}}) is isotopic to a standard homeomorphism g (defined as a map induced by a map (I_{n}, w_{p_{n}}) → (I_{n}, w_{p_{n}}) of inverse systems); maps f and g have the same topological entropy and if the entropy is positive, then they are semi-conjugate [Kwapisz 2001, Theorem 4, p. 271].

13. If e is the end point of B_{1} and S, T are composants of B_{1} which do not contain e, then there exists a continuous injection h : B_{1} \ {e} → B_{1} \ {e} such that h(S) = T [Aarts et al. 1991, Theorem 5.1].

Any two composants of B_{1} which do not contain e are homeomorphic [Bandt 1994].

14. Every autohomeomorphism of the buckethandle continuum has at least two fixed points [Aarts et al. 1998].

15. If A is a Borel subset of K ∈ K and A is the union of a family of composants of K, then A is meager or comeager [Emeryk 1980], [Krasinkiewicz 1974a].

16. Each composant C of B_{1} different from the one containing the end point is internal, i.e., every continuum L ⊂ ℝ^{2} intersecting both C and ℝ^{2} \ B_{1} must intersect all composants of B_{1} [Krasinkiewicz 1974b, Theorem, p. 261].

17. Any Knaster continuum K = \underleftarrow{lim}(I_{n}, f_{n}), where f_{m} = f_{n} for each m, n, is homeomorphic to the attracting set of a homeomorphism h : D^{2} → h(D^{2}) ⊂ D^{2} of the plane closed disk [Barge 1986]. If f_{n}(t) = 1 − |2t − 1| for all n (i.e., K = B_{1}), then h can be defined as a diffeomorphism which is called a horse-shoe map—it was first described by S. Smale in [Smale 1965] (see also [Barge 1988]). See Figure C.

18. The buckethandle B_{1} and all Knaster continua with two end points admit no mean [Illanes XXXX], [Kawamura et al. 1996, Theorem 2.2, p. 99].

19. The hyperspace C(K) of all subcontinua of any Knaster continuum K is homeomorphic to the cone Cone(K) over K by a homeomorphism h : C(K) → Cone(K) which sends K onto the vertex of Cone(K) and the set of all singletons of K onto the base of the cone [Dilks et al. 1981, Corollary 12, p. 639].

3.3 Unicoherent continua (also hereditary)

3.4 Discoherent continua

3.5 Multicoherent continua

Figure 3.2.1: ( C ) the horse-shoe map f maps the stadium shaped region D (A ∪ R ∪ B) in such a way, that it shrinks R vertically, stretches R horizontally, contracts the semicircles then folds the space once and places this result into itself so that f(A) and f(B) are in the interior of A and f(R) is in the interior of D.

IRREDUCIBLE CONTINUA

4.1 Elementary examples

(a) An arc, Sin Curve

(b) Other examples (Cantor Organ, Cantor Accordion, etc.)

4.1.1 Sin curve

The (topologist’s) sin curve S is defined by

S = \{ (x, y) ∈ ℝ^{2} : y = sin \frac{1}{x}, 0 < x ≤ 1 \} ∪ \{ (0, y) : −1 ≤ y ≤ 1 \}.

1. S is irreducible between points (0, y) and (1, sin 1), −1 ≤ y ≤ 1. It has exactly three end points and two arc components.

2. It is one of the simplest arc-like continua.

3. It is a compactification of a ray (0, 1] with remainder an arc.

4. It has the periodic-recurrent property [Charatonik et al. 1997b, Corollary 5.10, p. 117].

5. The only possible confluent nondegenerate images of S are an arc and a continuum homeomorphic to S [Nadler 1977a].

6. The hyperspace C(S) of all subcontinua of S is homeomorphic to the cone over S [Nadler 1977b].

See Figure A. There are many variations of the sin curve. Some of them are pictured below.

See Figure B–C.

46 CHAPTER 4. IRREDUCIBLE CONTINUA

Figure 4.1.1: ( A ) sin curve Figure 4.1.1: ( B ) union of two sin curves Figure 4.1.1: ( C ) union of two sin curves

Figure 4.1.2: ( A ) Cantor organ

Figure 4.1.2: ( B ) Cantor accordion

4.1.2 Cantor organ and accordion

The Cantor organ X is the union of the product C × I of the Cantor ternary set C and the unit interval I and all segments of the form J × {0} or J′ × {1}, where J (J′, resp.) is the closure of a component of I \ C of length 1/3^{2k−1} (1/3^{2k}), k ∈ ℕ [Kuratowski 1968, p. 191]. See Figure A.

1. X is an arc-like continuum which is irreducible between points (0, x) and (1, y), where 0 ≤ x, y ≤ 1, and has exactly four end points.

2. It has uncountably many arc components.

A variation of the Cantor organ is the Cantor accordion Y which is defined as the monotone image of X under a map that shrinks horizontal bars J × {0} and J′ × {1} to points [Kuratowski 1968, p. 191]. See Figure B.

Besides the above properties,

Figure 4.3.1: ( A ) dyadic selenoid

• Y has an upper semi-continuous monotone decomposition into arcs with the quotient space an arc.

4.2 Arc-like (chainable) continua

(a) The simplest arclike continua with n end points

(b) The pseudo-arc

4.3 Irreducible circle-like continua

(a) Solenoids

(b) The pseudo-circle, pseudo-solenoids

4.3.1 Solenoids

A solenoid is a continuum homeomorphic to the inverse limit Σ(p) = \underleftarrow{lim}(S^{1}, f_{n}) of the inverse sequence of unit circles S^{1} in the complex plane with bonding maps f_{n}(z) = z^{p_{n}}, where p = (p_{1}, p_{2}, …) is a sequence of prime numbers; it is called a p-adic solenoid . The solenoid Σ(2, 2, …) is known as a dyadic solenoid .

Geometrically, solenoid Σ(p) can be described as the intersection of a sequence of solid tori T_{1} ⊃ T_{2} ⊃ … such that T_{n+1} wraps p_{n} times around T_{n} without folding and T_{n} is ϵ_{n}-thin, for each n ∈ ℕ, where lim ϵ_{n} = 0. See Figure A.

1. Each solenoid can be constructed (up to a homeomorphism) as the quotient space of the product C × I by identifying each point (c, 1) with (h(c), 0), where h : C → C is a homeomorphism of the Cantor set C such that for every ϵ > 0 there exist a closed-open subset D of C and a positive integer n such that {f^{j}(D): j = 1, ⋯, n} is a cover of C consisting of pairwise disjoint subsets of C with diameters less than ϵ [Gutek 1980].

2. Each solenoid Σ(p) is an Abelian topological group with a group operation (z_{1}, z_{2}, …) · (z′_{1}, z′_{2}, …) = (z_{1} · z′_{1}, z_{2} · z′_{2}, …) and the neutral element e = (1, 1, …).

3. Either of the following conditions is equivalent for a nondegenerate continuum X different from a simple closed curve to be a solenoid.

(a) X is homeomorhic to a one-dimensional topological group [Hewitt 1963];

(b) X is indecomposable and is homeomorphic to a topological group [Chigogidze 1996, Theorem 8.6.18];

(c) X is circle-like, has the property of Kelley and contains no local end point [Krupski 1984c, Theorem (4.3)];

(d) X is circle-like, has the property of Kelley, each proper nondegenerate subcontinuum of X is an arc and X has no end pointsend point;

(e) X is circle-like, has the property of Kelley and has an open cover by Cantor bundles of open arcs (i.e., sets homeomorphic to the product C × (0, 1) of the Cantor set C and the open interval (0, 1)) [Krupski 1982];

(f) X is homogeneous, contains no proper, nondegenerate, terminal subcontinua and sufficiently small subcontinua of X are not ∞-ods [Krupski 1995, Theorem 3.1];

(g) X is a homogeneous curve containing an open subset U such that some component of U does not have the disjoint arcs property [Krupski 1995, p. 166];

(h) X is a homogeneous finitely cyclic (or, equivalently, k-junctioned) curve that is not tree-like and contains no nondegenerate, proper, terminal subcontinua [Krupski et al. XXXXb], [Duda et al. 1991].

(i) X is openly homogeneous and sufficiently small subcontinua of X are arcs [Prajs 1989];

4. Solenoid Σ(q) is a continuous image of Σ(p) if and only if the sequence q = (q_{1}, q_{2}, …) is a factorant of sequence p = (p_{1}, p_{2}, …), i.e., there exists i such that for each j ≥ i there is k such that q_{i} · · · · · q_{j} is a factor of p_{1} · · · · · p_{k}.

Two solenoids are homeomorphic if and only if each of them is a continuous image of another [Cook 1967], [D. van Dantzig 1930, Satz 8, p. 122].

There is a family of solenoids of cardinality 2^{ℵ_{0}} such that no member of the family is a continuous image of another.

5. Each monotonemonotone map image of a solenoid X is homeomorphic to X [Krupski 1984b, Theorem 5]. Each open map transforms X onto a solenoid or onto an arc-like continuum with the property of Kelley and with arcs as proper nondegenerate subcontinua; if the map is a local homeomorphism, then its image is a solenoid [Krupski 1984a].

6. The composant of a solenoid Σ containing e is a one-parameter topological subgroup of Σ, i.e. it is a one-to-one continuous homomorphic image of the additive group of the reals.

7. Any two composants of any two solenoids are homeomorphic [R. de Man 1995].

8. No solenoid can be mapped onto a strongly self-entwined continuum. In particular, it cannot be mapped onto a circle-like plane continuum which is a common part of a descending sequence of circular chains C_{i} such that C_{i+1} circles n times in C_{i} clockwisely and then n − 1 times counterclockwisely and the first link of C_{i} contains the closure of the first link of C_{i+1} [Rogers 1971b].

9. No movable continuum (in particular no continuum lying in a surface or a tree-like continuum) can be continuously mapped onto a solenoid. Alternatively, if the first Alexander-Čech cohomology group of a continuum X is finitely divisiblefinitely divisible group, then X cannot be mapped onto a solenoid [Krasinkiewicz 1976, Remark, p. 46, 4.1, 4.9., 5.1], [Krasinkiewicz 1978, Corollary 7.3], [Rogers 1975].

10. Every nonplanar, circle-like continuum has the shape of a solenoid [Krasinkiewicz 1976, remark, p. 46]. Two solenoids have the same shape if and only if they are homeomorphic [Godlewski 1970].

11. Any autohomeomorphism f of Σ(p) is isotopic to a homeomorphism g which is induced by a map (S^{1}, f_{n}) → (S^{1}, f_{n}) of the inverse sequences which define Σ(p) (g can be a group translation, the involution, a power map or its inverse, or compositions of these maps). Maps f and g have equal the topological entropies and are semi-conjugate if the entropy is positive [Kwapisz 2001, Theorems 1–3, pp. 252–253], [D. van Dantzig 1930, Satz 9, p. 125]. The topological group of all autohomeomorphisms (with the compactopen topology) of a solenoid Σ is homeomorphic (but not isomorphic) to the the product Σ × l_{2} × Aut(Σ), where l_{2} is the Hilbert space and the group Aut(Σ) of all topological group automorphisms of Σ is equipped with the discrete topology and it is equal to ℤ_{2}, or ℤ_{2} × ℤ^{n}, or ℤ_{2} ⊕_{i=1}^{∞} ℤ [Keesling 1972, Theorems 3.1 and 2.4].

12. If the spaces of all autohomeomorphisms of two solenoids are homeomorphic, then the solenoids are isomorphic as topological groups [Keesling 1972, Corollary 3.9].

4.4. COMPACTIFICATIONS OF THE REAL LINE AND THE REAL HALF-LINE51

13. Any map f : Σ(p) → Σ(q) is, for every ϵ > 0, ϵ-homotopic to a map induced by a map (S^{1}, f_{n}) → (S^{1}, g_{n}) between inverse sequences defining the corresponding solenoids [Rogers et al. 1971].

14. A p-adic solenoid admits a mean if and only if infinitely many numbers in the sequence p equal 2 [Krupski XXXXa]. The same condition is equivalent to the non-existence of exactly 2-to-1 map defined on the solenoid [Dȩbski 1992].

15. The hyperspace of all subcontinua of any solenoid Σ is homeomorphic the cone over Σ [Rogers 1971a], [Nadler 1991, p. 202].

16. The family of all solenoids in the cube I^{3} (as a subset of the hyperspace C(I^{3})) is Borel and not G_{δσ} [Krupski XXXXc].

4.4 Compactifications of the real line and the real half-line

(a) Waraszkiewicz spirals

4.5 Other examples

52 CHAPTER 4. IRREDUCIBLE CONTINUA

USC DECOMPOSITIONS OF CONTINUA

5.1 Decomposition spaces 5.2 Upper semi-continuous decompositions 5.3 Continuous decompositions

54 CHAPTER 5. USC DECOMPOSITIONS OF CONTINUA

CURVES

6.1 Order of a space at a point

(a) Curves of a finite order

(b) Regular continua

(c) Rational continua

(d) Suslinian continua

(e) Urysohn examples X(n), X(ℵ_{0}); X(2) = 𝕊^{1}

6.2 Case-Chamberlin examples

6.3 Curves without arcs

6.4 Covering defined classes (inverse limit ex-

pansions)

(a) Arc-like continua

(b) Circle-like continua (the circle, the Warsaw circle and other elementary examples; the circle of pseudo-arcs)

(c) Tree-like continua (dendrites, dendroids, λ-dendroids; Ingram’s example, Bellamy’s example)

6.5 Other examples

56 CHAPTER 6. CURVES

FAMILIES OF CONTNINUA

7.1 Universal elements

(a) Universal arc-like continuum

(b) Universal circle-like continuum

(c) Universal hereditarily indecomposable continuum

(d) Universal P-compacta

(e) Universal smooth dendroids

(f) Universal smooth fan

(g) Sierpiński universal plane curve

(h) Menger universal curve

7.2 Common models

7.3 Families of incomparable continua

58 CHAPTER 7. FAMILIES OF CONTNINUA

PLANE CONTINUA

8.1 Embeddability of continua in surfaces

8.2 Planability in general

8.3 Planability of graphs on of locally connect-

ed continua

(a) Kuratowski’s graphs

(b) Claytor’s examples of nonplanable curves

8.4 Separation of the plane

(a) Common boundary of plane domains

(b) Kuratowski’s example

(c) Hereditary cut of the plane (Whyburn’s curve)

60 CHAPTER 8. PLANE CONTINUA

Chapter 9

MAPPINGS

9.1 Various kinds of mappings and relations between them

62 CHAPTER 9. MAPPINGS

CONTINUA WITH SPECIAL MAPPING PROPERTIES

10.1 Existence of special mappings

(a) Retractions from hyperspaces

(b) Means

(c) Contractibility (also hereditary)

(d) Selectibility

10.2 Self-similar continua 10.3 Self-homeomorphic continua 10.4 Rigid continua

(a) Cook’s continua 64CHAPTER 10. CONTINUA WITH SPECIAL MAPPING PROPERTIES

10.5 Chaotic continua 10.6 Absolute retracts; spaces LC^{n} and LC^{∞} 10.7 Absolute retracts for various classes of

continua 10.8 Class (W) and other classes defined by

mappings 10.9 Homogeneity 10.10 Variations of homogeneity 10.11 Generalized homogeneity 10.12 Embeddability in ℝ^{n} 10.13 Weakly chainable continua 10.14 Fixed point property

Chapter 11

CONTINUA AS FRACTALS

66 CHAPTER 11. CONTINUA AS FRACTALS

Chapter 12

DYNAMICAL SYSTEMS

12.1 Attractors

68 CHAPTER 12. DYNAMICAL SYSTEMS

OTHER PROPERTIES

13.1 Colocal connectedness 13.2 Aposyndesis and its variations 13.3 Property of Kelley (also hereditary) 13.4 Smoothness 13.5 Arc-smoothness 13.6 C^{∗}-smoothness

70 CHAPTER 13. OTHER PROPERTIES

NEW FIGURES

14.1 New examples 1

14.1.1 Whyburn’s curve W

Whyburn’s curve

14.1.2 Pseudo-arc

Pseudo-arc

14.1.3 Zig-zag circle

zig-zag circle

14.1.4 M-continuum

M-continuum

14.1.5 Cantor interaction

Cantor interaction

14.1.6 Pseudo-circle

pseudo-circle

14.1.7 Semicircle continuum

semicircle continuum 72 CHAPTER 14. NEW FIGURES

Figure 14.1.1: ( A ) Whyburn’s curve Figure 14.1.2: ( A ) Pseudo-arc Figure 14.1.2: ( B ) Pseudo-arc

Figure 14.1.2: ( C ) Pseudo-arc 74 CHAPTER 14. NEW FIGURES

Figure 14.1.2: ( D ) Pseudo-arc Figure 14.1.2: ( E ) Pseudo-arc 14.1. NEW EXAMPLES 1 75

Figure 14.1.2: ( F ) Pseudo-arc 76 CHAPTER 14. NEW FIGURES

Figure 14.1.3: ( A ) zig-zag circle Figure 14.1.4: ( A ) M-continuum 14.1. NEW EXAMPLES 1 77

Figure 14.1.5: ( A ) Cantor interaction 78 CHAPTER 14. NEW FIGURES

Figure 14.1.6: ( A ) pseudo-circle 14.1. NEW EXAMPLES 1 79

Figure 14.1.7: ( A ) semicircle continuum Figure 14.1.8: ( A ) boxes continuum

Figure 14.2.1: ( A ) Graph K_{3,3}

14.1.8 Boxes continuum

boxes continuum

14.2 New examples 2

14.2.1 Graph K_{3,3}

Graph K_{3,3}

14.2.2 Graph K_{5}

Graph K_{5}

Figure 14.2.2: ( A ) Graph K_{5}

Figure 14.2.3: ( A ) space filling curve

14.2.3 Space filling curve

space filling curve

14.2.4 Universal arc-like continuum

universal arc-like continuum

14.2.5 Lakes of Vada

lakes of Vada

14.2.6 Borsuk fan

Borsul fan 82 CHAPTER 14. NEW FIGURES

Figure 14.2.4: ( A ) universal arc-like continuum 14.2. NEW EXAMPLES 2 83

Figure 14.2.5: ( A ) lakes of Vada

Figure 14.2.6: ( A ) Borsuk fan

14.2.7 Ingram’s atriodic continuum

Ingram’s atriodic continuum

14.2.8 Double Warsaw circle

double Warsaw circle

14.2.9 Warsaw circle

Warsaw circle

14.2.10 Crooked Warsaw circle

crooked Warsaw circle

14.3 New examples 3

14.3.1 Young spiral

Young spiral

14.3.2 Young space

Young space 14.3. NEW EXAMPLES 3 85

Figure 14.2.7: ( A ) Ingram’s atriodic continuum 86 CHAPTER 14. NEW FIGURES

Figure 14.2.8: ( A ) double Warsaw circle Figure 14.2.9: ( A ) Warsaw circle Figure 14.2.10: ( A ) crooked Warsaw circle 14.3. NEW EXAMPLES 3 87

Figure 14.3.1: ( A ) Young spiral 88 CHAPTER 14. NEW FIGURES

Figure 14.3.2: ( A ) Young space

Figure 14.3.3: ( A ) Bing space

14.3.3 Bing space

Bing space

14.3.4 Universal smooth dendroid

universal smooth dendroid

14.3.5 Claytor’s curve A

Claytor’s curve A

14.3.6 Claytor’s curve B

Claytor’s curve B

14.3.7 Sin addiction continuum

sin addiction continuum 90 CHAPTER 14. NEW FIGURES

Figure 14.3.4: ( A ) universal smooth dendroid Figure 14.3.5: ( A ) Claytor’s curve A 14.3. NEW EXAMPLES 3 91

Figure 14.3.6: ( A ) Claytor’s curve B Figure 14.3.7: ( A ) sin addiction continuum Figure 14.3.8: ( A ) dumbbell

Figure 14.3.9: ( A ) figure eight

14.3.8 Dumbbell

dumbbell

14.3.9 Figure Eight

figure eight

14.3.10 Noose

noose

Figure 14.3.10: ( A ) noose

Figure 14.4.1: ( A ) theta curve

Figure 14.4.2: ( A ) trapezoid basins continuum

14.4 New examples 4

14.4.1 Theta curve

theta curve

14.4.2 Trapezoid basins continuum

trapezoid basins continuum

14.4.3 Dragon continuum

dragon continuum

L. Fearnley, D. G. Wright : Geometric realization of a Bellamy Continuum, BUll. London Math. Soc. 15(1993), 177-183.

94 CHAPTER 14. NEW FIGURES Figure 14.4.3: ( A ) dragon continuum 14.4. NEW EXAMPLES 4 95

Figure 14.4.3: ( B ) dragon continuum

Figure 14.5.1: ( A ) harmonic fan

14.5 New examples 5

14.5.1 Harmonic fan

harmonic fan

14.5.2 Harmonic shredded fan

harmonic shredded fan

14.5.3 Cantor fan

Cantor fan

14.5.4 Lelek fan

Lelek fan 14.5. NEW EXAMPLES 5 97

Figure 14.5.2: ( A ) harmonic shredded fan Figure 14.5.3: ( A ) Cantor fan

Figure 14.5.4: ( A ) Lelek fan

14.6 New examples 6

14.6.1 Sierpiński triangle

Sierpiński triangle

14.6.2 Makuchowski umbrella

Makuchowski umbrella

14.6.3 Snake

snake

14.6.4 Cantor snake

Cantor snake

14.6.5 Cruller

cruller

14.7 New examples 7

14.7.1 Cantor comb

Cantor comb

14.7.2 Gehman dendrite on ski

Gehman dendrite on ski 14.7. NEW EXAMPLES 7 99

Figure 14.6.1: ( A ) Sierpiński triangle

Figure 14.6.2: ( A ) Makuchowski umbrella 100 CHAPTER 14. NEW FIGURES

Figure 14.6.3: ( A ) snake Figure 14.6.4: ( A ) Cantor snake 14.7. NEW EXAMPLES 7 101

Figure 14.6.5: ( A ) cruller Figure 14.7.1: ( A ) Cantor comb Figure 14.7.2: ( A ) Gehman dendrite on ski

Figure 14.7.3: ( A ) Makuchowski twin

14.7.3 Makuchowski twin

Makuchowski twin

14.7.4 Cantor function comb

Cantor function comb

14.7.5 Oversteegen’s dendroid

Oversteegen’s dendroid

Lex G. Oversteegen, Open retractions and locally confluent mappings of certain continua, Houston J. Math. 6,1 (1980), 113–125.

14.7. NEW EXAMPLES 7 103 Figure 14.7.4: ( A ) Cantor function comb 104 CHAPTER 14. NEW FIGURES

Figure 14.7.5: ( A ) Oversteegen’s dendroid 14.7. NEW EXAMPLES 7 105 Figure 14.7.5: ( B ) Oversteegen’s dendroid - a bunch of Cantor function combs Figure 14.7.5: ( C ) Oversteegen’s dendroid - from above (observe that the Cantor function combs have teeths shifted using the z^{2} function, starting from left to right)

Appendix A

Definitions

absolute end point

A point p of a continuum X is called an absolute end point of X if X \ {p} is a composant of X.

absolute retract

An absolute retract is a space X such that if X is embedded as a closed subset X′ of a space Y, then X′ is a retract of Y.

absolutely terminal continuum

A proper subcontinuum K of a continuum X is said to be an absolutely terminal continuum of X provided that K is a terminal continuum of each subcontinuum L of X which properly contains K.

acyclic

A continuum X is said to be acyclic provided that each mapping from X into the unit circle 𝕊^{1} is homotopic to a constant mapping, i.e., for all mappings f : X → 𝕊^{1} and c : X → {p} ⊂ 𝕊^{1} there exists a mapping h : X × [0, 1] → 𝕊^{1} such that for each point x ∈ X we have h(x, 0) = f(x) and h(x, 1) = c(x).

almost chainable

A continuum M is almost chainable if, for each positive number ε, there exists an open cover D of M with mesh ε and a chain C of elements of D and an end link L of C such that no element of D \ C intersects any link of C other than L and every point of M is at a distance less than ε from some link of C.

aposyndetic

A connected space X is aposyndetic at H with respect to K if there is a closed connected subset of X with H in its interior and not intersecting K, and X is aposyndetic if it is aposyndetic at each point with respect to every other point.

A continuum X is said to be aposyndetic provided that for each point p ∈ X and for each q ∈ X \ {p} there exists a subcontinuum K of X and an open set U of X such that p ∈ U ⊂ K ⊂ X \ {q} (see e.g.[Nadler 1992, Exercise 1.22, p. 12]).

arc

An arc is any space which is homeomorphic to the closed interval [0, 1].

arc-smooth

Given a continuum X with an arc-structure A, the pair (X, A) (see arc-structure) is said to be arc-smooth at a point v ∈ X provided that the induced function A_{v} : X → C(X) defined by A_{v}(x) = A(v, x) is continuous. Then the point v is called an initial point of (X, A). The pair (X, A) is said to be arc-smooth provided that there exists a point in X at which (X, A) is arc-smooth. An arbitrary space X is said to be arc-smooth at a point v ∈ X provided that there exists an arc-structure A on X for which (X, A) is arc-smooth at v. The space X is said to be arc-smooth if it is arc-smooth at some point (see [Fugate et al. 1981, p. 546]). Note that a dendroid is smooth if and only if it is arc-smooth.

arc-structure

By an arc-structure on an arbitrary space X we understand a function A : X × X → C(X) such that for every two distinct points x and y in X the set A(x, y) is an arc from x to y and that the following metric-like axioms are satisfied for every points x, y and z in X:

(1) A(x, x) = {x};

(2) A(x, y) = A(y, x);

(3) A(x, z) ⊂ A(x, y) ∪ A(y, z), with equality prevailing whenever y ∈ A(x, z).

We put (X, A) to denote that the space X is equipped with an arc-structure A (see [Fugate et al. 1981, p. 546]). Note that if there exists an arc-structure on a continuum, then the continuum is arcwise connected.

atomic

A surjective mapping f : X → Y between compact spaces is said to be (see [Maćkowiak 1979, Chapter 3 and 4, p. 12-28]):

– atomic provided that for each subcontinuum K of X such that the set f(K) is nondegenerate we have K = f^{−1}(f(K));

atriodic

a continuum that contains no subcontinuum which is itself a triod is said to be atriodic

biconnected

A space is biconnected if it is not the sum of two mutually exclusive nondegenerate connected point sets.

bihomogeneous

Suppose X is a space such that, for each two points a and b, there is a homeomorphism of X onto itself such that h(a) = b and h(b) = a. Then X is bihomogeneous.

Borel sets

The family F of Borel sets of a space S is the smallest family satisfying the conditions:

(i) Every closed set belongs to F;

(ii) If X is in F, then S \ X is in F;

(iii) The countable intersection of elements of F belongs to F.

branch point

A branch point of a continuum is the vertex of a simple triod lying in that continuum.

chainable

A simple chain is a finite sequence L_{1}, L_{2}, ⋯, L_{n} of open sets such that L_{i} intersects L_{j} if and only if |i − j| ≤ 1. The terms of the sequence L_{1}, L_{2}, ⋯, L_{n} are called the links of the chain. An ε-chain is a chain each of whose links has diameter less than ε. a continuum M is chainable if, for each positive number ε, M can be covered by an ε-chain.

chaotic

A nondegenerate topological space X is said to be chaotic if for any two distinct points p and q of X there exist an open neighborhood U of p and an open neighborhood V of q such that no open subset of U is homeomorphic to any open subset of V.

circle

A circle is any space which is homeomorphic to the unit circle.

completely regular

a continuum is completely regular if each nondegenerate subcontinuum has a nonempty interior.

composant

If a point p of a continuum X is given, then the composant of X belonging to p is defined to be the union of all proper subcontinua of X which contain p. A set is a composant of X provided it is the composant of X belonging to p for some point p of X.

confluent

A mapping f of a compact space X onto a compact space Y is confluent if, for each continuum C in Y, each component of f^{−1}(C) is mapped onto C by f.

-connected

A topological space (X, τ) is said to be σ-connected provided that X is not the union of more than one and at most countably many nonempty, mutually disjoint, closed subsets.

A continuum is δ-connected if each two of its points can be connected by an hereditarily decomposable irreducible subcontinuum.

connected im kleinen

A space is connected im kleinen if it is aposyndetic at each point with respect to each closed point set not containing that point.

continuous selection of

Let a continuum X, a compact space Y and a function F : X → 2^{Y} be given. A function f : X → Y is called a continuous selection of F provided that it is continuous and f(x) ∈ F (x) for each x ∈ X.

converges homeomorphically to the continuum

The statement that the sequence M_{1}, M_{2}, ⋯ converges homeomorphically to the continuum M means there exists a sequence h_{1}, h_{2}, ⋯ of homeomorphisms such that, for each positive integer i, h_{i} is a homeomorphism from M_{i} onto M and for each positive number ε there exists a positive integer N such that if j > N then, for all x, dist(h_{j}(x), x) < ε.

converge 0-regularly

A sequence of sets {A_{n}} contained in a metric space X with a metric d is said to converge 0-regularly to its limit A = Lim A_{n} (see hyperspace ) provided that for each ε > 0 there is a δ > 0 and there is an index n_{0} ∈ ℕ such that if n > n_{0} then for every two points p, q ∈ A_{n} with d(p, q) < δ there is a connected set C_{n} ⊂ A_{n} satisfying conditions p, q ∈ C_{n} and diam_{n} < ε (see [Whyburn 1942, Chapter 9, §3, p. 174]).

convex

Given a continuum X with an arc-structure A, a subset Z of X is said to be convex provided that for each pair of points x and y of Z the arc A(x, y) is a subset of Z. If Z is a convex subcontinuum of X, then A|Z×Z is an arc-structure on Z. We define X to be locally convex at a point p ∈ X provided that for each open set U containing p there is a convex set Z such that p ∈ int Z ⊂ cl Z ⊂ U (see [Fugate et al. 1981, I.2, p. 548-549]).

continuum

A continuum is a nonempty, compact, connected metric space.

contractible

A continuum X is said to be contractible provided there is a homotopy h : X × [0, 1] → X such that for some point p ∈ X we have h(x, 0) = x and h(x, 1) = p for each x ∈ X (see e.g. [Nadler 1978, (16.2), p. 532]).

curve

A curve is any one-dimensional continuum.

cut point

Let S, T) be a connected topological space, and let p ∈ S. If S\{p} is connected, then p is called a non-cut point of S. If S \ {p} is not connected, then p is called a cut point of S.

C^{∗}-smooth

Let X be a continuum. Define C^{∗} : C(X) → C(C(X)) by C^{∗}(A) = C(A). It is known that for any continuum X the function C^{∗} is upper semi-continuous, [Nadler 1978, Theorem 15.2, p. 514], and it is continuous on a dense G_{δ} subset of C(X), [Nadler 1978, Corollary 15.3, p. 515]. A continuum X is said to be C^{∗}-smooth at A ∈ C(X) provided that the function C^{∗} is continuous at A. A continuum X is said to be C^{∗}-smooth provided that the function C^{∗} is continuous on C(X), i.e., at each A ∈ C(X) (see [Nadler 1978, Definition 5.15, p. 517]).

cyclic element

Let a continuum X be locally connected and let p ∈ X. By a cyclic element of X containing p we mean either the singleton {p} if p is a cut point or an end point of X, or the set consisting of p and of all points x ∈ X such that no point of X cuts X between p and x, otherwise (see e.g. [Whyburn 1942, Chapter 4, §2, p. 66]).

decomposable

a continuum X is said to be decomposable provided that X can be written as the union of two proper subcontinua.

deformation retract of

Let X be a space and let A and B be subspaces of X with A ⊂ B. Then A is called a deformation retract of B over X provided that the identity mapping i_{B} : B → B is homotopic in X to a retraction r : B → A. Further, A is called a strong deformation retract of B over X provided that it is a deformation retract of B over X and the homotopy keeps the points of A fixed throughout the entire deformation of B into A (see e.g. [Dugundji 1966, Definition 6.3, p. 324]).

dendroid

A dendroid is an arcwise connected, hereditarily unicoherent continuum.

dendrite

A dendrite means a locally connected continuum containing no simple closed curve.

dispersion point

A point p of a nondegenerate connected space X is an explosion point, or dispersion point, of X provided that X \ {p} is totally disconnected.

end point

Point of order 1 in a continuum X is called end point of X; the set of all end points of X is denoted by E(X).

equivalent

Let 𝔐 be a class of mappings. Two spaces X and Y are said to be equivalent with respect to 𝔐 (shortly 𝔐-equivalent) if there are two mappings, both in 𝔐, one from X onto Y and the other from Y onto X. If 𝔐 means the class of monotone mappings, we say that X and Y are monotonely equivalent.

F -equivalent

If X and Y are continua, we say that Y is F -equivalent to X provided there is a mapping in F from X onto Y and a mapping in F from Y onto X.

feebly monotone

Let X and Y be continua. A mapping f : X → Y is said to be feebly monotone provided that if A and B are proper subcontinua of Y such that Y = A ∪ B, then their inverse images f^{−1}(A) and f^{−1}(B) are connected.

finitely Suslinean

a continuum X is finitely Suslinean if, for each positive number ε, X does not contain infinitely many mutually exclusive subcontinua of diameter greater than ε.

finitely linear

A function f I → Y, where I is a closed interval of the real line, is called finitely linear provided there exists a positive integer m such that I can be decomposed, for each ε > 0, into a finite number of closed subintervals I_{1}, I_{2}, ⋯, I_{k} each of length less than ε and with the property that the set f(I_{i}) meets at most m of the sets f(I_{1}), f(I_{2}), ⋯, f(I_{k}) for i = 1, 2, ⋯, k.

fixed point

We say that a function F : X → 2^{X} (or a function F : X → C(X)) has a fixed point provided that there is a point x ∈ X such that x ∈ F (x).

fixed set property

A topological space X is said to have the fixed set property for a certain class C of maps of X onto itself provided there exists, for each non-empty closed set A in X, a map f in C such that f(x) = x if and only if x is in A.

fixed point self-homeomorphic

A topological space X is called fixed point self-homeomorphic if for any point p, any neighborhood U of p, there is an embedding h : X → U with h(p) = p and p ∈ int h(X).

fixed ball self-homeomorphic

A topological space X is called fixed ball self-homeomorphic if for any point p, any neighborhood U of p, there is a neighborhood V of p with V ⊆ U and an embedding h : X → U satisfying h|_{V} = id_{V}.

free arc

A free arc in X is any subset of X of the form A \ {x, y} where A is an arc in X with end points x and y and A \ {x, y} is open in X.

graph

A graph is a continuum which can be written as the union of finely many arcs any two of which are either disjoint or intersect only in one or both their end points.

half-ray curve

a continuum is a half-ray curve if it is a continuous 1-1 image of the nonnegative reals.

hereditary

A property of a continuum X is said to be hereditary if every subcontinuum of X has the property. In particular, a continuum is said to be hereditarily unicoherent if the intersection of any two of its subcontinua is connected.

hereditarily

Let 𝔐 be a class of mappings between continua. A mapping f : X → Y between continua is said to be hereditarily 𝔐 provided that its restriction to any subcontinuum of the domain X is in 𝔐 (see [Maćkowiak 1979, Chapter 4, Section B, p. 16]).

hereditarily decomposable

a continuum X is said to be hereditarily decomposable provided that each nondegenerate subcontinuum of X (as well as X itself) is decomposable.

hereditarily equivalent

a continuum is hereditarily equivalent if it is homeomorphic to each of its nondegenerate subcontinua.

hereditarily indecomposable

a continuum X is said to be hereditarily indecomposable provided that each of its subcontinua is indecomposable, that is, for each subcontinuum C ⊂ X and for every continua A and B such that A ∪ B = C we have either A = C or B = C.

hereditarily locally connected

A continuum X is said to be hereditarily locally connected, written hlc, provided that every subcontinuum of X is a Peano Continuum.

hereditarily unicoherent

A connected topological space is said to be hereditarily unicoherent provided that each of its closed, connected subsets is unicoherent.

homogeneous

Let 𝔐 be a class of mappings. A space X is said to be homogeneous with respect to 𝔐 (or shortly 𝔐-homogeneous) provided that for every two points p and q of X there is a surjective mapping f : X → X such that f(p) = q and f ∈ 𝔐. If 𝔐 is the class of homeomorphisms, we get the concept of a homogeneous space.

HU-terminal

A subcontinuum K of a hereditarily unicoherent continuum X is said to be a HU-terminal continuum of X provided that K is contained in an irreducible subcontinuum of X and for every irreducible subcontinuum I of X containing K there is a point x ∈ X such that I is irreducible about the union K ∪ {x}.

hyperspace

Given a continuum X with a metric d, we let 2^{X} to denote the hyperspace of all nonempty closed subsets of X equipped with the Hausdorff metric H defined by

H(A, B) = max{sup{d(a, B) : a ∈ A}, sup{d(b, A) : b ∈ B}} for A, B ∈ 2^{X}

(see e.g. [Nadler 1978, (0.1), p. 1 and (0.12), p. 10]). If H(A, A_{n}) tends to zero as n tends to infinity, we put A = Lim A_{n}. Further, we denote by F_{1}(X) the hyperspace of singletons of X, and by C(X) the hyperspace of all subcontinua of X, i.e., of all connected elements of 2^{X}. Since X is homeomorphic to F_{1}(X), there is a natural embedding of X into C(X), and so we can write X ⊂ C(X) ⊂ 2^{X}. Thus one can consider a retraction from either C(X) or 2^{X} onto X.

indecomposable

a continuum X is said to be indecomposable provided that X cannot be written as the union of two proper subcontinua.

induced mapping

Let f : X → Y be a mapping between continua. Then the induced mapping C(f) : C(X) → C(Y) is defined by C(f)(A) = f(A), where A in the left member of the equality means an element of C(X), while in the right one it is understood as a subcontinuum of X (see [Nadler 1978, (0.49), p. 23]).

inverse limit

The reader can find necessary information on the inverse limits of inverse sequences e.g. in Section 2 of the second chapter of [Nadler 1992], as well as in [Kuratowski 1968].

irreducible

Let X be a continuum and A ⊂ X. Then, X is said to be irreducible about A provided no proper subcontinuum of X contains A. a continuum X is said to be irreducible provided that X is irreducible about {p, q} for some p, q ∈ X.

light

A mapping is light if each point inverse is totally disconnected.

like

If X is a metric space, a mapping f from X to a space Y is an ε-map if, for each point y of Y, diamf^{−1}(y)) ≤ ε. If C is a collection of continua, a continuum M is C-like if, for every positive number ε, there exists an ε-map of M onto an element of C. In particular, a continuum is tree-like if, for some collection C of trees, M is C-like.

A concept of a tree-like continuum can be defined in several (equivalent) ways. One of them is the following. A continuum X is said to be tree-like provided that for each ε > 0 there is a tree T and a surjective mapping f : X → T such that f is an ε-mapping (i.e., diam^{−1}(y) < ε for each y ∈ T). Let us mention that a continuum X is tree-like if and only if it is the inverse limit of an inverse sequence of trees with surjective bonding mappings. Compare e.g. [Nadler 1992, p. 24].

Using a concept of a nerve of a covering, one can reformulate the above definition saying that a continuum X is be tree-like provided that for each ε > 0 there is an ε-covering of X whose nerve is a tree.

Finally, the original definition using tree-chains can be found e.g. in Bing’s paper [Bing 1951, p. 653].

locally

Let 𝔐 be a class of mappings between compact spaces. A surjective mapping f : X → Y between continua is said to be locally 𝔐 provided that for each point x ∈ X there is a closed neighborhood V of x such that f(V) is is a closed neighborhood of f(x) and that the restriction f|V is in 𝔐 (see [Maćkowiak 1979, Chapter 4, Section C, p. 18]).

locally confluent

A mapping f from a space X onto a space Y is locally confluent if, for each point y of Y, there is an open set U containing y such that f^{−1}( \overline{U} ) is confluent.

local homeomorphism

A surjective mapping f : X → Y between compact spaces is said to be (see [Maćkowiak 1979, Chapter 3 and 4, p. 12-28]):

– a local homeomorphism provided that for each point x ∈ X there exists a an open neighborhood U of x such that f(U) is an open neighborhood of f(x) and that f restricted to U is a homeomorphism between U and f(U).

local separating point

A point p of a locally compact separable metric space L is a local separating point of L provided there exists an open set U of L containing p and two points x and y of the component containing p of U such that U \ {p} is the sum of two mutually separated point sets, one containing x and the other containing y.

MO-mapping

A surjective mapping f : X → Y between compact spaces is said to be (see [Maćkowiak 1979, Chapter 3 and 4, p. 12-28]):

– an MO-mapping provided that it can be represented as the composition of two mappings, f = f_{2} ∘ f_{1} such that f_{1} is open and f_{2} is monotone.

monotone map

A continuous function f : X → Y is said to be monotone provided f^{−1}(y) is connected for each y ∈ Y.

mutually aposyndetic

a continuum M is mutually aposyndetic if, for each two points A and B of M, there exist mutually exclusive subcontinua H and K of M containing A and B, respectively, in their interiors. a continuum is strictly non-mutually aposyndetic if each two of its subcontinua with interiors intersect.

natural projection

Let continua X and Y be given. A mapping f : X ×Y → X is called the natural projection provided that it is defined by f((x, y)) = x.

neat

A dendroid X is said to be neat provided that each one of its subdendroids has no improper shore point (see [Neumann-Lara et al. 1993, p. 939]).

Let 𝔐 be a class of mappings. A class 𝔐 of mappings is said to be neat provided that if all homeomorphisms are in 𝔐 and the composition of any two mappings in 𝔐 is also in 𝔐.

near homeomorphism

A mapping f : X → Y is called a near homeomorphism provided that for each ε > 0 there is a homeomorphism h : X → Y such that sup{d(f(x), h(x)) : x ∈ X} < ε.

OM-mapping

A surjective mapping f : X → Y between compact spaces is said to be (see [Maćkowiak 1979, Chapter 3 and 4, p. 12-28]):

– an OM-mapping provided that it can be represented as the composition of two mappings, f = f_{2} ∘ f_{1} such that f_{1} is monotone and f_{2} is open.

open map

A mapping f of a space X into a space Y is open if, for each open A in X the image f(A) is open in Y.

openly minimal

We say that a dendrite X is openly minimal provided that every open image of X can be openly mapped onto X.

orbit

Given a space X let H(X) stand for the group of autohomeomorphisms of X. If a point p ∈ X is fixed, then {h(p) ∈ X : h ∈ H(X)} is called an orbit of p. Orbits of points of X either are mutually disjoint or coincide, and their union is the whole X.

order

Let (X, T) be a topological space, and let A ⊂ X. Let β be a cardinal number. We say that A is of order less than equal to β in X, written ord(A, X)≤ β, provided that for each U ∈ T such that A ⊂ U, there exists V ∈ T such that A ⊂ V ⊂ U and |Bd(V)| ≤ β.

We say that A is of order β in X, written ord(A, X)= β, provided that ord(A, X)≤ β and ord(A, X)≰ α for any cardinal number α < β.

A concept of an order of a point p in a continuum X (in the sense of Menger-Urysohn), written ord (p, X), is defined as follows. Let 𝔫 stand for a cardinal number. We write:

ord (p, X) ≤ 𝔫 provided that for every ε > 0 there is an open neighborhood U of p such that diam ≤ ε and card bd U ≤ 𝔫;

ord (p, X) = 𝔫 provided that ord (p, X) ≤ 𝔫 and for each cardinal number 𝔪 < 𝔫 the condition ord (p, X) ≤ 𝔪 does not hold;

ord (p, X) = ω provided that the point p has arbitrarily small open neighborhoods U with finite boundaries bd U and card bd U is not bounded by any n ∈ ℕ.

Thus, for any continuum X we have

ord (p, X) ∈ {1, 2, …, n, …, ω, ℵ_{0}, 2^{ℵ_{0}}}

(convention: ω < ℵ_{0}); see [Kuratowski 1968, §51, I, p. 274].

Let a dendroid X and a point p ∈ X be given. Then p is said to be a point of order at least 𝔪 in the classical sense provided that p is the center of an 𝔪-od contained in X. We say that p is a point of order 𝔪 in the classical sense provided that 𝔪 is the minimum cardinality for which the above condition is satisfied (see [Charatonik 1962, p. 229]).

order preserving mapping

For each point p of a continuum X equipped with an arc-structure A : X ×X → C(X) we define a partial order ≤_{p} by letting x ≤_{p} y whenever A(p, x) ⊂ A(p, y). Let X and Y be continua with fixed arc-structures A and B, respectively. We say that a surjective mapping f : X → Y is a ≤_{p}-mapping provided that x ≤_{p} y in X implies that f(x) ≤_{f(p)} f(y) in Y. If, in addition, Y ⊂ X, B = A|(Y × Y), and f is a retraction, then f is called a ≤_{p}-retraction (or ≤_{p}-preserving retraction). The concept of a <_{p}-mapping is defined in a similar manner (with f(x) ≠ f(y) implied by x ≠ y). For order preserving mappings see e.g. [Fugate et al. 1981, I.7, p. 553].

ordinary point

Point of order 2 is called ordinary point of X; the set of all ordinary points of X is denoted by O(X).

periodic point of

Let a continuum X and a mapping f : X → X be given. For each natural number n denote by f^{n} the n-th iteration of f. A point p ∈ X is called a periodic point of f provided that there is n ∈ ℕ such that f^{n}(p) = p The set of periodic points of a mapping f : X → X are denoted by P (f).

periodic-recurrent property

The set of periodic points and of recurrent points of a mapping f : X → X are denoted by P (f) and R(f) respectively. Clearly P (f) ⊂ R(f). A continuum X is said to have the periodic-recurrent property (shortly PR-property) provided that for every mapping f : X → X the equality cl P (f) = cl R(f) holds.

pointwise self-homeomorphic

A topological space X is called pointwise self-homeomorphic at a point x ∈ X if for any neighborhood U of x there is a set V such that x ∈ V ⊆ U and V is homeomorphic to X. The space X is called pointwise self-homeomorphic if it is pointwise self-homeomorphic at each of its points.

pseudo-arc

A pseudo-arc is the only (up to homeomorphism) hereditarily indecomposable arc-like continuum.

pseudo-confluent

A mapping from X to Y is pseudo-confluent if every irreducible continuum in Y is the image of a continuum in X.

property of Kelley

A continuum X is said to have the property of Kelley provided that for each point x ∈ X, for each subcontinuum K of X containing x and for each sequence of points x_{n} converging to x there exists a sequence of subcontinua K_{n} of X containing x_{n} and converging to the continuum K (see e.g. [Nadler 1978, Definition 16.10, p. 538]).

quasi-monotone

A surjective mapping f : X → Y between compact spaces is said to be (see [Maćkowiak 1979, Chapter 3 and 4, p. 12-28]) quasi-monotone provided that for each subcontinuum Q of Y with the nonempty interior the set f^{−1}(Q) has a finite number of components and f maps each of them onto Q.

radially convex at

Given a continuum X with an arc-structure A and a point p ∈ X, a metric d on X is said to be radially convex at p provided that d(p, z) = d(p, y) + d(y, z) for every points y, z ∈ X with y ∈ A(p, z) (see [Fugate et al. 1981, I.4, p. 551]).

ramification point

Point of order at least 3 is called ramification point of X; the set of all ramification points of X is denoted by R(X).

rational

A continuum is rational if every two of its points can be separated by a countable point set.

real curve

A continuum is a real curve if it is a continuous 1-1 image of the real line.

recurrent point of

Let a continuum X and a mapping f : X → X be given. For each natural number n denote by f^{n} the n-th iteration of f. A point p ∈ X is called a recurrent point of f provided that for every neighborhood U of p there is n ∈ ℕ such that f^{n}(p) ∈ U. The set of recurrent points of a mapping f : X → X are denoted by R(f).

refinable (monotonely)

A map r from a compact metric space X onto a compact metric space Y is (monotonely) refinable if, for each positive number ε, there is a (monotone) ε-map f from X onto Y such that, for each x in X,

d (f (x) , r (x)) < ε.

regular

If X is a continuum and p ∈ X, then X is said to be regular at p provided that there is a local base L_{p} at p such that the boundary of each member of L_{p} is of finite cardinality. A continuum is said to be regular provided that X is regular at each of its points.

A continuum is regular if each two of its points can be separated by a finite point set.

retract, retraction

Let X and Y be continua. A mapping f : X → Y is said to be a retraction provided that Y ⊂ X and the restriction f|Y : Y → f(Y) ⊂ X is the identity; then Y is called a retract of X.

rigid

A nondegenerate topological space X is said to be rigid if it has a trivial autohomeomorphism group, i.e., if the only homeomorphism of X onto X is the identity.

selectible

A continuum X is said to be selectible provided that there exists a mapping σ : C(X) → X (called a selection for C(X)) such that σ(A) ∈ A for each continuum A ⊂ X (see e.g. [Nadler 1978, p. 253]).

scattered

A point set X is scattered if every subset Y of X has a point that is not a limit point of Y.

self-homeomorphic

A topological space X is called self-homeomorphic if for any open set U ⊆ X there is a set V ⊆ U such that V is homeomorphic to X.

semi-aposyndetic

A space X is semi-aposyndetic if, for each two of its points, X is aposyndetic at one of them with respect to the other.

semi-confluent

A surjective mapping f : X → Y between compact spaces is said to be (see [Maćkowiak 1979, Chapter 3 and 4, p. 12-28]):

– semi-confluent provided that for each subcontinuum Q of Y and for every two components C_{1} and C_{2} of f^{−1}(Q) either f(C_{1}) ⊂ f(C_{2}) or f(C_{2}) ⊂ f(C_{1}).

semi-continuum

A semi-continuum is a continuumwise connected point set.

semi-continuous

Let a continuum X, a compact space Y and a function F : X → 2^{Y} be given. Put

F^{−1}(B) = {x ∈ X : F (x) ∩ B ≠ ∅}.

The function F is said to be lower (upper) semi-continuous provided that F^{−1}(B) is open (closed) for each open (closed) subset B ⊂ Y. It is said to be continuous provided that it is both lower and upper semi-continuous. This notion of continuity agrees with the one for mappings between metric spaces.

semi-locally connected

A continuum M is semi-locally connected at the point p if, for every domain D containing p, there exits a domain E lying in D and containing p such that M \E has only a finite number of components. A continuum is totally non-semi-locally connected if it is not semi-locally connected at any point.

simple closed curve

A simple closed curve is any space which is homeomorphic to the unit circle.

simple 𝔪-od

Let 𝔪 be a cardinal number. By a simple 𝔪-od with the center p we mean the union of 𝔪 arcs every two of which have p as the only common point.

simple triod

The union of three arcs emanating from a point v is called a simple triod provided the singleton v is the intersection of any two of the arcs ; the point v is then called the top of the simple triod, and the arcs are called its arms.

shore point

A point p of a dendroid X is called a shore point of X if there exists a sequence of subdendroids X_{n} of X \ {p} such that X = Lim X_{n}. A shore point of X that is not an end point of X is called an improper shore point of X (see [Neumann-Lara et al. 1993, p. 939]).

smooth (dendroid)

A dendroid X is said to be smooth at a point p ∈ X provided that for each point x ∈ X and for each sequence of points {x_{n}} in X tending to x, the sequence of arcs px_{n} tends to the arc px. A dendroid X is said to be smooth provided that it is smooth at some point p ∈ X (see [Charatonik et al. 1970, p. 298]).

strongly chaotic

A nondegenerate topological space X is said to be strongly chaotic if for any two distinct points p and q of X there exist open neighborhoods U of p and V of q respectively such that no open subset of U is homeomorphic to any subset of V;

strongly pointwise self-homeomorphic

A topological space X is called strongly pointwise self-homeomorphic at a point x ∈ X if for any neighborhood U of x there is a neighborhood V of x such that x ∈ V ⊆ U and V is homeomorphic to X. The space X is called strongly pointwise self-homeomorphic if it is strongly pointwise self-homeomorphic at each of its points.

strongly rigid

A nondegenerate topological space X is said to be strongly rigid if the only homeomorphism of X into X is the identity of X onto itself.

strongly self-homeomorphic

A topological space X is called strongly self-homeomorphic if for any open set U ⊆ X there is a set V ⊆ U with nonempty interior such that V is homeomorphic to X.

strongly unicoherent1

A continuum X is strongly unicoherent provided X is unicoherent and each proper subcontinuum with interior is unicoherent.

strongly unicoherent2

A continuum X is said to be strongly unicoherent2 provided that it is unicoherent and for each pair of its proper subcontinua A and B such that X = A ∪ B, each of A and B is unicoherent (see [Bennett 1971, p. 587]).

Suslinean

A continuum is Suslinean if it does not contain uncountably many mutually exclusive nondegenerate subcontinua.

T_{A}(x)

For each point x of a continuum X with an arc-structure A we define (see [Fugate et al. 1981, I.3, p. 550]) T_{A}(x) = {y ∈ X : each convex subcontinuum of X with y in its interior contains x}. Since T_{A}(x) is always closed, we have T_{A} : X → 2^{X}.

terminal continuum

A proper subcontinuum K of a continuum X is said to be a terminal continuum of X provided that if whenever A and B are proper subcontinua of X having union equal to X such that A ∩ K ≠ ∅ ≠ B ∩ K, then either X = A ∪ K or X = B ∪ K.

thin

A topological space S is thin if and only if for each two homeomorphic subset A and B of S, there is a homeomorphism h of S onto itself such that h(A) = B.

translation

If P and Q are subspaces of a metric space X with a metric d, and ε is a positive number, then a mapping g : P → Q is called an ε-translation provided that d(p, g(p)) < ε for each point p ∈ P.

tree

A tree is a graph which contains no simple closed curve. A tree as a onedimensional acyclic connected polyhedron, i.e., a dendrite with finitely many end points.

triod

A continuum is called a triod provided it contains a subcontinuum whose complement is the union of three nonempty pairwise disjoint open sets.

unicoherent

A connected topological space S is said to be unicoherent provided that whenever A and B are closed, connected subsets of S such that S = A ∪ B, then A ∩ B is connected.

Let a continuum X and its subcontinuum Y be given. Then X is said to be unicoherent at Y provided that for each pair of proper subcontinua A and B of X such that X = A ∪ B the intersection A ∩ B ∩ Y is connected (see [Owens 1986, p. 146]).

uniformly continuum-chainable

A continuum X is said to be uniformly continuum-chainable if for each positive number ε there is an integer k = k(ε) such that for each pair x,y of points of X, there are subcontinua A_{1}, ⋯, A_{k} of X each of diameter less than ε such that x ∈ A_{1}, y ∈ A_{k} and A_{i} ∩ A_{j} ≠ ∅ whenewer |i − j| ≤ 1.

uniquely arcwise connected

A continuum is said to be uniquely arcwise connected provided that for every two of its points there is exactly one arc in the continuum joining these points.

universal

Let a class S of spaces be given. A member U of S is said to be universal for S if every member of S can be embedded in U, i.e., if for every X ∈ S there exists a homeomorphism h : X → h(X) ⊂ U. Accordingly, a dendrite is said to be universal if it contains a homeomorphic image of any other dendrite.

weak cut point

A point x of a continuum M is a weak cut point of M if there are two points p and q of M such that every subcontinuum of M that contains both p and q also contains x.

weakly chainable

A continuum is weakly chainable if it is a continuous image of a chainable continuum.

weakly confluent

A surjective mapping f : X → Y between compact spaces is said to be (see [Maćkowiak 1979, Chapter 3 and 4, p. 12-28]) weakly confluent provided that for each subcontinuum Q of Y there is a component the set f^{−1}(Q) which is mapped under f onto Q.

weakly hereditarily unicoherent

A continuum is said to be weakly hereditarily unicoherent if the intersection of any two of its subcontinua with nonempty interiors is connected (see [Owens 1986, p. 152] and references therein).

weakly monotone

A surjective mapping f : X → Y between compact spaces is said to be (see [Maćkowiak 1979, Chapter 3 and 4, p. 12-28]) weakly monotone provided that for each subcontinuum Q of Y with the nonempty interior each component the set f^{−1}(Q) is mapped under f onto Q.

weakly smooth at a point

A dendroid X is said to be weakly smooth at a point p ∈ X provided that the subspace of 2^{X} consisting of all subarcs of X of the form px for x ∈ X is compact (see [Lum 1975, p. 113]).

weakly Suslinian

A continuum X is said to be weakly Suslinian provided that X is not the union of more than one pairwise disjoint nondegenerate subcontinua of X.

widely connected

A nondegenerate connected space X is widely connected if each nondegenerate connected subset of X is dense in X.

width w(X)

For any compact metric space X, the width w(X) of X is the l.u.b. of the set of all real numbers a which satisfy the following condition: for each ε > 0, there exists a finite open cover C of X such that meshC) < ε and for each chain C′ which is a subcollection of C there is a member A of C such that d(A, C′^{∗}) ≥ a.

130 APPENDIX A. DEFINITIONS

Appendix B

List of examples

Graph K_{3,3}, p. 80 Bing space, p. 89 Borsul fan, p. 81 boxes continuum, p. 80 buckethandle, p. 39 Cantor accordion, p. 47 Cantor comb, p. 98 Cantor fan, p. 96 Cantor function comb, p. 102 Cantor interaction, p. 71 Cantor organ, p. 47 Cantor snake, p. 98 Claytor’s curve A, p. 89 Claytor’s curve B, p. 89 comb W, p. 11 comb W_{R}, p. 11 crooked Warsaw circle, p. 84 cruller, p. 98 dendrite D_{m}, p. 13 dendrite D_{S}, p. 17 dendrite L_{0}, p. 22 dendrite of de Groot-Wille type, p. 25 dendrite P, p. 18 double buckethandle, p. 39 double Warsaw circle, p. 84 dragon continuum, p. 93 dumbbell, p. 92 dyadic solenoid, p. 48 figure eight, p. 92 Gehman dendrite, p. 27 Gehman dendrite on ski, p. 98 Graph K_{5}, p. 80 harmonic fan, p. 96 harmonic shredded fan, p. 96 horseshoe, p. 39 Ingram’s atriodic continuum, p. 84 Knaster continuum, p. 39 lakes of Vada, p. 81 Lelek fan, p. 96 locally connected combs, p. 11 locally connected fan, p. 11 Makuchowski twin, p. 102 Makuchowski umbrella, p. 98 M-continuum, p. 71 Menger universal curve, p. 34 Miller dendrite S, p. 24 modified Miller dendrite, p. 26 noose, p. 92 dendrite G_{ω}, p. 28 Omiljanowski dendrite, p. 19 Oversteegen’s dendroid, p. 102 p-adic Knaster continuum, p. 41 p-adic solenoid, p. 48 Pseudo-arc, p. 71 pseudo-circle, p. 71 semicircle continuum, p. 71 Sierpiński carpet, p. 31 Sierpiński triangle, p. 98 Sierpiński universal plane curve, p. 31 sin addiction continuum, p. 89 sin curve, p. 45 snake, p. 98 solenoid, p. 48 space filling curve, p. 81 standard universal dendrite of order m, p. 13 standard universal dendrite of orders in S, p. 17 theta curve, p. 93 topologist’s sin curve, p. 45 trapezoid basins continuum, p. 93 universal arc-like continuum, p. 81 universal smooth dendroid, p. 89 Warsaw circle, p. 84 Ważewski universal dendrite, p. 12 Whyburn’s curve, p. 71 Young space, p. 84 Young spiral, p. 84 zig-zag circle, p. 71 133

Index

-connected, 110 -equivalent, 113 -chain, 110 -like, 117 -map, 117 -od, 125 -smooth, 112 -smooth at, 112

absolute end point, 107 absolute retract, 107 absolutely terminal continuum, 107 acyclic, 107 almost chainable, 107 aposyndetic, 108 aposyndetic at, 108 arc, 108 arc-smooth, 108 arc-smooth at a point, 108 arc-structure, 108 arms, 125 atomic, 109 atriodic, 109

biconnected, 109 bihomogeneous, 109 Bing space, 89 Borel sets, 109 Borsul fan, 81 boxes continuum, 80 branch point, 109 buckethandle, 39

Cantor accordion, 47 Cantor comb, 98 Cantor fan, 96 Cantor function comb, 102 Cantor interaction, 71 Cantor organ, 47 Cantor snake, 98 circle, 110 Claytor’s curve A, 89 Claytor’s curve B, 89 comb W, 11 comb W_{R}, 11 completely regular, 110 composant of, 110 composant of X belonging to, 110 confluent, 110 connected im kleinen, 111 continuous, 124 continuous selection of, 111 continuum, 111 contractible, 112 converge 0-regularly, 111 converges homeomorphically to the

continuum, 111 convex, 111 crooked Warsaw circle, 84 cruller, 98 curve, 112 cut point of, 112 cyclic element, 112

decomposable, 112 deformation retract of, 113 dendrite, 113 dendrite D_{m}, 13 dendrite D_{S}, 17 dendrite G_{ω}, 28 dendrite L_{0}, 22 dendrite of de Groot-Wille type, 25 dendrite P, 18 dendroid, 113 dispersion point, 113 double buckethandle, 39 double Warsaw circle, 84 dragon continuum, 93 dumbbell, 92 dyadic solenoid, 48

equivalent, 113

monotonely, 113 explosion point, 113

figure eight, 92 finitely linear, 114 finitely Suslinean, 114 fixed point, 114 fixed set property, 114 free arc, 115

Gehman dendrite, 27 Gehman dendrite on ski, 98 graph, 115 Graph K_{3,3}, 80 Graph K_{5}, 80

half-ray curve, 115 harmonic fan, 96 harmonic shredded fan, 96 Hausdorff metric, 116 hereditarily, 115 hereditarily decomposable, 115 hereditarily equivalent, 115 hereditarily indecomposable, 116 hereditarily locally connected, 116 hereditarily unicoherent, 116 hereditary, 115 hlc, 116 homeomorphism

near, 119 homogeneous, 116

with respect to, 116 horse-shoe map, 43 horseshoe, 39 HU-terminal continuum, 116 hyperspace, 116

chainable, 110 chaotic, 110

strongly, 125

improper shore point, 125 indecomposable, 117 induced mapping, 117 Ingram’s atriodic continuum, 84 initial point, 108 irreducible, 117 irreducible about, 117 is of order less than equal to, 120

Knaster continuum, 39

lakes of Vada, 81 Lelek fan, 96 light, 117 links, 110 local homeomorphism, 118 local separating point, 118 locally, 118 locally confluent, 118 locally connected combs, 11 locally connected fan, 11 locally convex at a point, 111

M-continuum, 71

Makuchowski twin, 102 Makuchowski umbrella, 98 Menger universal curve, 34 Miller dendrite S, 24 minimal

openly, 120 MO-mapping, 119 modified Miller dendrite, 26 monotone

feebly, 114 monotone, 119 mutually aposyndetic, 119

natural projection, 119

neat, 119

non-cut point of, 112

noose, 92

OM-mapping, 119

Omiljanowski dendrite, 19

open, 120 orbit of, 120 order, 120 order preserving mapping, 121 Oversteegen’s dendroid, 102

point end, 113 ordinary, 121 periodic, 121 ramification, 122 recurent, 123 weak cut, 128

PR-property, 121 preserving retraction, 121 property

periodic-recurrent, 121 property of Kelley, 122 pseudo-arc, 121 Pseudo-arc, 71 pseudo-circle, 71 pseudo-confluent, 122

quasi-monotone, 122

radially convex at, 122 rational, 122 real curve, 122 refinable, 123 regular, 123 regular at, 123 retract, 123 retraction, 123 rigid, 123

strongly, 126

scattered, 124 selectible, 123 selection, 123 self-homeomorphic, 124

fixed ball, 114 point, 114

pointwise, 121 at a point, 121 strongly, 125

strongly, 126 semi-aposyndetic, 124 semi-confluent, 124 semi-continuous, 124 semi-continuum, 124 semi-locally connected, 124 semicircle continuum, 71 shore point, 125 Sierpiński carpet, 31 Sierpiński triangle, 98 Sierpiński universal plane curve, 31 simple closed curve, 125 simple chain, 110 simple triod, 125 sin addiction continuum, 89 sin curve, 45 smooth, 125 smooth at a point, 125 snake, 98 solenoid, 48 space filling curve, 81 standard universal dendrite of order

m, 13 standard universal dendrite of or-

ders in S, 17 strictly non-mutually aposyndetic,

119 strong deformation retract of, 113 strongly unicoherent, 126 strongly unicoherent2, 126 Suslinean, 126

terminal continuum, 126 theta curve, 93 thin, 127 top, 125 topologist’s sin curve, 45 totally non-semi-locally connected,

124 translation, 127 trapezoid basins continuum, 93 tree, 127 tree-like, 117 triod, 127

unicoherent, 127 at, 127 uniformly continuum-chainable, 127 uniquely arcwise connected, 127 universal, 128

for, 128 universal arc-like continuum, 81 universal smooth dendroid, 89

Warsaw circle, 84 Ważewski universal dendrite, 12 weakly confluent, 128 weakly hereditarily unicoherent,

128 weakly chainable, 128 weakly monotone, 128 weakly smooth at a point, 128 weakly Suslinian, 129 Whyburn’s curve, 71 widely connected, 129 width w(X), 129

Young space, 84 Young spiral, 84

zig-zag circle, 71 138 INDEX

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