Examples in Continuum Theory

Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih February 21, 2001

2

Contents

1 CONNECTEDNESS 5

1.1 Connectedness, arcwise connectedness, continuumwise connect-

edness (continuum chainability) . . . . . . . . . . . . . . . . . . . 5 1.2 A cut point (the complement is not connected); a separating

point (the complement is not continuumwise connected) . . . . . 5 1.3 Components, quasi-components, composants. Arc-components. 5

2 LOCAL CONNECTEDNESS 7

2.1 Local connectedness and related concepts (connectedness im kleinen, aposyndesis, semi-local connectedness); relations between them. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 DIMENSION 9 3.1 The concept of a curve. Mapping of [0, 1] onto [0, 1]^{2}. . . . . . . 9 3.2 Menger-Urysohn dimension of a space at a point. . . . . . . . . 9 3.3 Universal spaces (in general). . . . . . . . . . . . . . . . . . . . . 9

3.3.1 Hilbert’s cube . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3.2 Sierpiński Universal Plane Curve . . . . . . . . . . . . . 9 3.3.3 Menger’s universal curve. Its Anderson’s characterization. 10

3.4 Universal Menger’s compacta. Their Bestvina’s characterization. 10 3.5 Nöbeling’s examples. (?). . . . . . . . . . . . . . . . . . . . . . . 12 3.6 Embeddability of a curve in a surface. Planability. Two Ku-

ratowski’s graphs and two Claytor’s curves. Embeddability of a locally connected continuum in the plane. . . . . . . . . . . . . . 12

4 ORDER OF A POINT 13 4.1 Menger-Urysohn’s order of a point. . . . . . . . . . . . . . . . . 13 4.2 Curves of a finite order. . . . . . . . . . . . . . . . . . . . . . . . 13 4.3 Urysohn’s examples: for each positive integer n ≥ 2 there is a

continuum X(n) consisting solely of points of order n and 2n − 2. 13 4.3.1 Sierpiński Tringle . . . . . . . . . . . . . . . . . . . . . . 13

4.4 Urysohn’s examples: for 𝔯 ∈ {ω, ℵ_{0}, 𝔠} there is a continuum X(𝔯) consisting solely of points of order 𝔯 . . . . . . . . . . . . . . . . 14 5 DENDRITES 15

5.1 Dendrites and their basic properties. . . . . . . . . . . . . . . . . 15 5.2 Universal dendrites of order n ≥ 3. Various constructions. … 15 5.3 Ważewski’s universal dendrite. . . . . . . . . . . . . . . . . . . . 15 5.4 Local dendrites. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 INDECOMPOSABILITY 17 6.1 Early examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.2 Common boundary of three or more domains. Examples by R.

L. Wilder and M. Lubański. . . . . . . . . . . . . . . . . . . . . 17 6.3 Indecomposable continua via inverse limits. . . . . . . . . . . . 17

7 IRREDUCIBLE CONTINUA; DECOMPOSITIONS 19 7.1 ??? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8 SPECIAL CONTINUA 21 8.1 Hereditarily decomposable curves without arcs (Z. Janiszewski). 21 8.2 Whyburn’s hereditary cut of the plane. . . . . . . . . . . . . . . 21 8.3 Hereditarily indecomposable continua. . . . . . . . . . . . . . . 21

8.3.1 The pseudoarc . . . . . . . . . . . . . . . . . . . . . . . . 21 8.3.2 The pseudocircle . . . . . . . . . . . . . . . . . . . . . . . 21 8.3.3 n-dimensional hereditarily indecomposable continua (R.H.Bing) 21

8.4 The circle of pseudoarcs. . . . . . . . . . . . . . . . . . . . . . . 21 8.5 P-like continua. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

9 UNIVERSAL SPACES 23 9.1 Universal arc-like continuum(Schori). . . . . . . . . . . . . . . . 23 9.2 Universal circle-like continuum(Rogers). . . . . . . . . . . . . . . 23 9.3 Some universal curves (Krasinkiewicz and Minc). . . . . . . . . . 23 9.4 Mohler-Nikiel universal smooth dendroid. . . . . . . . . . . . . . 23

10 MAPPINGS 25 10.1 Mapping properties — families of continua. . . . . . . . . . . . 25 10.2 Various kinds of mappings. . . . . . . . . . . . . . . . . . . . . 25 10.3 Fixed point theory. . . . . . . . . . . . . . . . . . . . . . . . . . 25

11 HOMOGENEOUS SPACES 27 11.1 Homogeneity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 11.2 Homogeneous continua. Solenoids. . . . . . . . . . . . . . . . . 27 11.3 Homogeneity of Menger compacta (Anderson, W. Lewis). . . . . 27 11.4 Variants of homogeneity; Knaster problem; K. Kuperberg’s results. 27 11.5 Generalized homogeneity. . . . . . . . . . . . . . . . . . . . . . 27

12 HYPERSPACES 29 12.1 Hyperspaces are terrible ;-) . . . . . . . . . . . . . . . . . . . . . 29 13 Lost continua 31

13.1 Arc-like continua . . . . . . . . . . . . . . . . . . . . . . . . . . 31 13.1.1 Topologist’s Sin Curve . . . . . . . . . . . . . . . . . . . 31 13.1.2 Open Sin . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 13.1.3 Double Topologists’s Sin Curve . . . . . . . . . . . . . . 32 13.1.4 Snake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 13.1.5 Cantor Meanders . . . . . . . . . . . . . . . . . . . . . . 33 13.1.6 Cantor Snake . . . . . . . . . . . . . . . . . . . . . . . . 33 13.1.7 Cruller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 13.1.8 Double Buckethandle . . . . . . . . . . . . . . . . . . . . 34

13.2 Self-homeomorphic continua . . . . . . . . . . . . . . . . . . . . 36

13.2.1 Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 13.3 Dendrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

13.3.1 Ważewski’s Universal Dendrite . . . . . . . . . . . . . . 37 13.3.2 Universal Dendrite D_{3} . . . . . . . . . . . . . . . . . . . 37 13.3.3 Universal Dendrite D_{4} . . . . . . . . . . . . . . . . . . . 37 13.3.4 Universal Dendrite D_{6} . . . . . . . . . . . . . . . . . . . 40 13.3.5 Gehman Dendrite . . . . . . . . . . . . . . . . . . . . . . 40

13.4 Dendroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 13.4.1 Cantor Comb . . . . . . . . . . . . . . . . . . . . . . . . 41 13.4.2 Gehman Dendrite with Ski . . . . . . . . . . . . . . . . . 41 13.4.3 Makuchowski Twin . . . . . . . . . . . . . . . . . . . . . 42

13.5 Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 13.5.1 Harmonic Fan . . . . . . . . . . . . . . . . . . . . . . . . 42 13.5.2 Harmonic Shredded Fan . . . . . . . . . . . . . . . . . . 43 13.5.3 Cantor Fan . . . . . . . . . . . . . . . . . . . . . . . . . . 43 13.5.4 Lelek Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

13.6 Planable continua . . . . . . . . . . . . . . . . . . . . . . . . . . 45 13.6.1 Makuchowski Umbrella . . . . . . . . . . . . . . . . . . . 45

14 New examples 47

14.1 New examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 14.1.1 Whyburn’s Curve . . . . . . . . . . . . . . . . . . . . . . 47 14.1.2 Pseudo-arc . . . . . . . . . . . . . . . . . . . . . . . . . . 50 14.1.3 Dyadic Selenoid . . . . . . . . . . . . . . . . . . . . . . . 52 14.1.4 Zig-zag Circle . . . . . . . . . . . . . . . . . . . . . . . . 56 14.1.5 M-Continuum . . . . . . . . . . . . . . . . . . . . . . . . 56 14.1.6 Buckethandle . . . . . . . . . . . . . . . . . . . . . . . . 56 14.1.7 Cantor Interaction . . . . . . . . . . . . . . . . . . . . . . 58 14.1.8 Pseudo-circle . . . . . . . . . . . . . . . . . . . . . . . . . 58 14.1.9 Semicircle Continuum . . . . . . . . . . . . . . . . . . . . 58 14.1.10 Boxes Continuum . . . . . . . . . . . . . . . . . . . . . . 59

14.2 New examples 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 14.2.1 Graph K_{3,3} . . . . . . . . . . . . . . . . . . . . . . . . . . 64 14.2.2 Graph K_{5} . . . . . . . . . . . . . . . . . . . . . . . . . . 64 14.2.3 Space Filling Curve . . . . . . . . . . . . . . . . . . . . . 64 14.2.4 Universal Arc-like Continuum . . . . . . . . . . . . . . . 66 14.2.5 Lakes of Wada . . . . . . . . . . . . . . . . . . . . . . . . 66 14.2.6 Borsuk Fan . . . . . . . . . . . . . . . . . . . . . . . . . . 66 14.2.7 Ingram’s Atriodic Continuum . . . . . . . . . . . . . . . 66 14.2.8 Double Warsaw Circle . . . . . . . . . . . . . . . . . . . . 66 14.2.9 Warsaw Circle . . . . . . . . . . . . . . . . . . . . . . . . 66 14.2.10 Crooked Warsaw Circle . . . . . . . . . . . . . . . . . . . 66

14.3 New examples 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 14.3.1 Young Spiral . . . . . . . . . . . . . . . . . . . . . . . . . 69 14.3.2 Young Space . . . . . . . . . . . . . . . . . . . . . . . . . 69 14.3.3 Bing Space . . . . . . . . . . . . . . . . . . . . . . . . . . 69 14.3.4 Universal Smooth Dendroid . . . . . . . . . . . . . . . . 69 14.3.5 Claytor’s Curve A . . . . . . . . . . . . . . . . . . . . . . 69 14.3.6 Claytor’s Curve B . . . . . . . . . . . . . . . . . . . . . . 69 14.3.7 Sin Addiction Continuum . . . . . . . . . . . . . . . . . . 76 14.3.8 Dumbbell . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 14.3.9 Figure Eight . . . . . . . . . . . . . . . . . . . . . . . . . 76 14.3.10 Noose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

14.4 New examples 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 14.4.1 Theta Curve . . . . . . . . . . . . . . . . . . . . . . . . . 77 14.4.2 Trapezoid Basins Continuum . . . . . . . . . . . . . . . . 77 14.4.3 Dragon Continuum . . . . . . . . . . . . . . . . . . . . . 77

A Definitions 81

B List of examples 103

Chapter 1

CONNECTEDNESS

Last section contains lost continua.

1.1 Connectedness, arcwise connectedness, continuumwise connectedness (continuum chainability)

The arc-like continua are beautiful

1.2 A cut point (the complement is not connected); a separating point (the complement is not continuumwise connected)

The arc-like continua are beautiful

1.3 Components, quasi-components, composants.

Arc-components.

The arc-like continua are beautiful 8 CHAPTER 1. CONNECTEDNESS

LOCAL CONNECTEDNESS

Last section contains lost continua.

2.1 Local connectedness and related concepts (connectedness im kleinen, aposyndesis, semi-local connectedness); relations between them.

The arc-like continua are beautiful 10 CHAPTER 2. LOCAL CONNECTEDNESS

DIMENSION

Last section contains lost continua.

3.1 The concept of a curve. Mapping of [0, 1] onto [0, 1]^{2}.

The arc-like continua are beautiful

3.2 Menger-Urysohn dimension of a space at a point.

The arc-like continua are beautiful

3.3 Universal spaces (in general).

The arc-like continua are beautiful

3.3.1 Hilbert’s cube

3.3.2 Sierpiński Universal Plane Curve

The Sierpiński Universal Plane Curve is a well known continuum which serves as the universal element in the class of all one-dimensional continua in the plane. It is obtained as the residual set remaining when one begins with a square and applies the operation of dividing it into nine equal squares and omitting the interior of the center one, then repeats this operation on each of the surviving 8 squares, then repeats again on the surviving 64 squares, and so on · · · . See [65, p.9].

The Sierpiński Universal Plane Curve can be characterized as the only plane locally connected one-dimensional continuum S such that the boundary of each

Figure 3.3.2: ( A ) Sierpiński Universal Plane Curve

complementary domain of S is a simple closed curve and no two of these complementary domain boundaries intersect. See [79].

3.3.3 Menger’s universal curve. Its Anderson’s characterization.

The Menger Universal Curve X is obtained using the Sierpiński Universal Plane Curve S from the unit cube I^{3} by the formula

X = {(x, y, z) ∈ I^{3} : (x, y) ∈ S & (x, z) ∈ S & (y, z) ∈ S} .

3.4 Universal Menger’s compacta. Their Bestvina’s characterization.

The arc-like continua are beautiful 3.4. UNIVERSAL MENGER’S COMPACTA. THEIR BESTVINA’S CHARACTERIZATION. 13 Figure 3.3.3: ( A ) Menger’s universal curve. Its Anderson’s characterization.

3.5 Nöbeling’s examples. (?).

The arc-like continua are beautiful

3.6 Embeddability of a curve in a surface. Planability. Two Kuratowski’s graphs and two Claytor’s curves. Embeddability of a locally connected continuum in the plane.

The arc-like continua are beautiful

ORDER OF A POINT

Last section contains lost continua.

4.1 Menger-Urysohn’s order of a point.

The arc-like continua are beautiful

4.2 Curves of a finite order.

The arc-like continua are beautiful

4.3 Urysohn’s examples: for each positive integer n ≥ 2 there is a continuum X(n) consisting solely of points of order n and 2n − 2.

The arc-like continua are beautiful

4.3.1 Sierpiński Tringle

The Sierpiński Triangle is obtained as the residual set remaining when one begins with a triangle and applies the operation of dividing it into four equal triangles and omitting the interior of the center one, then repeats this operation on each of the surviving 3 triangles, then repeats again on the surviving 9 triangles, and so on · · · . See [36, Example 2.7].

The Sierpiński Triangle is homeomorphic to the unique nonempty compact set K of the complex plane that satisfies

K = w_{1}(K) ∪ w_{2}(K) ∪ w_{3}(K) ,

where w_{1}, w_{2}, w_{3} are maps of the complex plane defined by w_{1}(z) = z/2, w_{2} = (z + 1)/2 and w_{3} = (z + i)/2.

16 CHAPTER 4. ORDER OF A POINT

Figure 4.3.1: ( A ) Sierpiński Tringle

4.4 Urysohn’s examples: for 𝔯 ∈ {ω, ℵ_{0}, 𝔠} there is a continuum X(𝔯) consisting solely of points of order 𝔯

The arc-like continua are beautiful

DENDRITES

Last section contains lost continua.

5.1 Dendrites and their basic properties.

The arc-like continua are beautiful

5.2 Universal dendrites of order n ≥ 3. Various

constructions.

The arc-like continua are beautiful

5.3 Ważewski’s universal dendrite.

The arc-like continua are beautiful

5.4 Local dendrites.

The arc-like continua are beautiful 18 CHAPTER 5. DENDRITES

INDECOMPOSABILITY

Last section contains lost continua.

6.1 Early examples.

The arc-like continua are beautiful

6.2 Common boundary of three or more domains. Examples by R. L. Wilder and M. Lubański.

The arc-like continua are beautiful

6.3 Indecomposable continua via inverse limits.

The arc-like continua are beautiful 20 CHAPTER 6. INDECOMPOSABILITY

IRREDUCIBLE CONTINUA; DECOMPOSITIONS

Last section contains lost continua.

7.1 ???

The arc-like continua are beautiful 22 CHAPTER 7. IRREDUCIBLE CONTINUA; DECOMPOSITIONS

SPECIAL CONTINUA

Last section contains lost continua.

8.1 Hereditarily decomposable curves without

arcs (Z. Janiszewski).

The arc-like continua are beautiful

8.2 Whyburn’s hereditary cut of the plane.

The arc-like continua are beautiful

8.3 Hereditarily indecomposable continua.

The arc-like continua are beautiful

8.3.1 The pseudoarc 8.3.2 The pseudocircle 8.3.3 n-dimensional hereditarily indecomposable contin-

ua (R.H.Bing)

8.4 The circle of pseudoarcs.

The arc-like continua are beautiful

8.5 P-like continua.

The arc-like continua are beautiful 24 CHAPTER 8. SPECIAL CONTINUA

UNIVERSAL SPACES

Last section contains lost continua.

9.1 Universal arc-like continuum(Schori).

The arc-like continua are beautiful

9.2 Universal circle-like continuum(Rogers).

The arc-like continua are beautiful

9.3 Some universal curves (Krasinkiewicz and

Minc).

The arc-like continua are beautiful

9.4 Mohler-Nikiel universal smooth dendroid.

The arc-like continua are beautiful 26 CHAPTER 9. UNIVERSAL SPACES

MAPPINGS

Last section contains lost continua.

10.1 Mapping properties — families of contin-

ua.

The arc-like continua are beautiful

10.2 Various kinds of mappings.

The arc-like continua are beautiful

10.3 Fixed point theory.

The arc-like continua are beautiful 28 CHAPTER 10. MAPPINGS

HOMOGENEOUS SPACES

Last section contains lost continua.

11.1 Homogeneity.

The arc-like continua are beautiful

11.2 Homogeneous continua. Solenoids.

The arc-like continua are beautiful

11.3 Homogeneity of Menger compacta (An-

derson, W. Lewis).

The arc-like continua are beautiful

11.4 Variants of homogeneity; Knaster prob-

lem; K. Kuperberg’s results.

The arc-like continua are beautiful

11.5 Generalized homogeneity.

The arc-like continua are beautiful 30 CHAPTER 11. HOMOGENEOUS SPACES

Chapter 12

HYPERSPACES

Last section contains lost continua.

12.1 Hyperspaces are terrible ;-)

The arc-like continua are beautiful 32 CHAPTER 12. HYPERSPACES

Lost continua

This section contains continua from the the first version.

13.1 Arc-like continua

The arc-like continua are beautiful

13.1.1 Topologist’s Sin Curve

The Topologist’s Sin Curve is well known continuum which is not arcwise connected. The continuum is defined by the union

{(x, y) ∈ R^{2} : 0 < x ≤ 1, y = sin(1/x)} ∪ {(0, y) ∈ R^{2} : −1 ≤ y ≤ 1} . It has just two path components. More we can find in [74, p.137] .

13.1.2 Open Sin

The Open Sin is defined by the Euclidean closure of the set {(x, y) ∈ R^{2} : −1 < x < 1, y = sin(1/(x^{2} − 1))} .

Figure 13.1.1: ( A ) Topologist’s Sin Curve

Figure 13.1.2: ( A ) Open Sin

Figure 13.1.3: ( A ) Double Topologists’s Sin Curve

This continuum has just three path components.

13.1.3 Double Topologists’s Sin Curve

The Double Topologists’s Sin Curve is defined by the union

{(x, y) ∈ R^{2} : 0 < |x| ≤ 1, y = sin(1/x)} ∪ {(0, y) ∈ R^{2} : −1 ≤ y ≤ 1} . It has just three path components.

13.1.4 Snake

The Snake is an arclike continuum where each open mapping is a homeomorphism.

We can find more in [40, Theorem ..].

Figure 13.1.4: ( A ) Snake Figure 13.1.5: ( A ) Cantor Meanders

Figure 13.1.6: ( A ) Cantor Snake

13.1.5 Cantor Meanders

The Cantor Meanders X is formed by combining the product C × I of the Cantor Ternary Set C and the unit interval I with the appropriate segments of I × {0, 1}. We use the ’zig-zag’ system of selecting the intervals. We start with [1/3, 2/3] × {0}, continue with [1/9, 2/9] × {1} and [7/9, 8/9] × {1}, · · · .

The space X has uncountably many arc components.

13.1.6 Cantor Snake

The Cantor Snake X is formed by combining the product C × I of the Cantor Ternary Set C and the unit interval I with countably many Open Sin Curves in such a way that the space X has uncountably many arc components.

Figure 13.1.7: ( A ) Cruller

13.1.7 Cruller

The Cruller continuum is beautiful.

13.1.8 Double Buckethandle

The Cantor 1/5-discontinuum C_{5} is created using the procedure of deleting the second and the fourth segments from five equal segments (instead of the middle third for the Cantor Ternary Set). C_{5} is the set of the reals in the unit interval which can be represented without 1-s and 3-s in the form

∑_{n=1}^{∞} \frac{a_{n}}{5^{n}} .

The Double Buckethandle continuum is the union of:

(i) the half circles in the lower plane going through points of C_{5} ∩ {x : 2/5^{n+1} ≤ x ≤ 1/5^{n}} with the centre 7/10 · 5^{n},

(ii) the half circles in the upper plane going through points of C_{5} ∩ {x : 2/5^{n+1} ≤ 1 − x ≤ 1/5^{n}} with the centre 1 − 7/10 · 5^{n}.

We can find more in [55, p.205 - p.206].

13.1. ARC-LIKE CONTINUA 37 Figure 13.1.8: ( A ) Double Buckethandle Figure 13.1.8: ( B ) Double Buckethandle

Figure 13.2.1: ( A ) Root

13.2 Self-homeomorphic continua

Self-homeomorphic continua are beautiful

13.2.1 Root

The Root is the continuum obtained attaching small ’T ’ letters to the previous ones as in the figure. It is a strongly pointwise self-homeomorphic dendrite. It has points of order three on the horizontal segments and points of order four on the vertical segments.

See [36, p.232-233].

Figure 13.3.1: ( A ) construction of the Ważewski’s Universal Dendrite

13.3 Dendrites

Dendrites are beautiful

13.3.1 Ważewski’s Universal Dendrite

The Ważewski’s Universal Dendrite is well known continuum which serves as the universal element in the class of all dendrites. We start with a Locally Connected Fan F_{ω}. Denote D_{1} = F_{ω}. Now we attach sufficiently small copies of F_{ω} to all midpoints of all maximal free arcs in D_{1} and obtain a dendrite D_{2}. Continuing in this fashion, we obtain the sequence of dendrites {D_{n}} and the Ważewski’s Universal Dendrite D is the closure of their union. For details see [65, p.181-185].

13.3.2 Universal Dendrite D_{3}

The Universal Dendrite D_{3} is obtained by the same procedure as the Ważewski’s Universal Dendrite just instead of locally connected fans we use simple triods.

We start with a simple triod T. Denote T_{1} = T. Now we attach sufficiently small copies of T (with one end point as the attaching point) to all midpoints of all maximal free arcs in T_{1} and obtain a dendrite T_{2}. Continuing in this fashion, we obtain the sequence of dendrites {T_{n}} and the Universal Dendrite 𝔻_{3} is the closure of their union.

The Universal Dendrite D_{3} is homogeneous with respect to the monotone mappings. For details see [38, p.362].

13.3.3 Universal Dendrite D_{4}

The Universal Dendrite D_{4} is a dendtrite with ramification points of order 4.

40 CHAPTER 13. LOST CONTINUA Figure 13.3.2: ( A ) Universal Dendrite D_{3} 13.3. DENDRITES 41

Figure 13.3.3: ( A ) Universal Dendrite D_{4}

Figure 13.3.4: ( A ) Universal Dendrite D_{6}

13.3.4 Universal Dendrite D_{6}

Universal Dendrite D_{6} is nice.

13.3.5 Gehman Dendrite

The Gehman Dendrite G is formed from the Cantor Ternary Set C in such a way that C is the set of end points of G and all ramification points of G are of order three.

See [67, pp. 422-423] and [68, p. 203].

13.4 Dendroids

Dendroids are beautiful Figure 13.3.5: ( A ) Gehman Dendrite

Figure 13.4.1: ( A ) Cantor Comb

13.4.1 Cantor Comb

The Cantor Comb X is formed from the Cantor Ternary Set C and the unit interval I using the formula

X = C × I ∪ I × {0} .

13.4.2 Gehman Dendrite with Ski

The Gehman Dendrite with Ski G_{s} is a dendroid formed from the Gehman Dendrite G and the Cantor Ternary Set C with a formula G_{s} = {(x, y, z) ∈ R^{3} : (x, y) ∈ G} ∪ {(x, y, z) ∈ R^{3} : x ∈ C, y = 0, |z| ≤ 1} . In fact we attach uncountably many segments of length 2 with their mid-

points to the points of Cantor Ternary Set (the end points of Gehman Dendrite). See [25, p. 16].

Figure 13.4.2: ( A ) Gehman Dendrite with Ski

Figure 13.4.3: ( A ) Makuchowski Twin

13.4.3 Makuchowski Twin

We obtain the Makuchowski Twin as a subset of the Euclidean plane with the induced topology by attaching the Locally Connected Fan to the end point of the prolonged segment of the Prolonged Harmonic Fan.

Makuchowski twin is a dendroid where each nonconstant open mapping is light. See [60, Example 2.1].

13.5 Fans

Fans are beautiful

13.5.1 Harmonic Fan

We define the Harmonic Fan F_{H} = ∪{ve : e ∈ H}, where ve is the segment joining v = (0, 1) with e ∈ H, H = {0} ∪ {1/n : n ∈ N}. We consider F_{H} as a

Figure 13.5.1: ( A ) Harmonic Fan

subset of the Euclidean plane with the induced topology.

See [39, p.9].

13.5.2 Harmonic Shredded Fan

To define the Harmonic Shredded Fan F_{HS} we cut the ’arms’ ve in the Harmonic Fan F_{H} in such a way, that their lengths run through all rationals in the unit interval (we do not cut the vertical segment through the origin). We notice that all Harmonic shredded fans are homeomorphic.

See [39, p.31].

13.5.3 Cantor Fan

We define the Cantor Fan F_{C} = ∪{ve : e ∈ C}, where ve is the segment joining v = (1/2, 1) with all points e in the Cantor Ternary Set C. We consider F_{C} as a subset of the Euclidean plane with the induced topology.

See [39, p.9].

46 CHAPTER 13. LOST CONTINUA

Figure 13.5.2: ( A ) Harmonic Shredded Fan Figure 13.5.3: ( A ) Cantor Fan

Figure 13.5.4: ( A ) Lelek Fan

13.5.4 Lelek Fan

We define the Lelek Fan F_{L} a subfan of the Cantor Fan F_{C} such that the set E(F_{L}) of its end points is dense in F_{L}, and S(F_{L}) = {v} ∪ E(F_{L}) is a connected set (where v denotes the top of the two fans F_{C} and F_{L} ⊂ F_{C}).

It is known, that any two Lelek Fans are homeomorphic. See [39, p.9] . Moreover the Lelek Fan is homeomorphic to all its confluent images (see [37, ]).

13.6 Planable continua

Planable continua are beautiful

13.6.1 Makuchowski Umbrella

Makuchowski Umbrella is a hereditarily locally connected continuum where each nonconstant open mapping is light. This property disappears when we cut out those vertical segments due to the horizontal projection on appropriate vertical segment.

See [60, Example 2.5].

48 CHAPTER 13. LOST CONTINUA Figure 13.6.1: ( A ) Makuchowski Umbrella

New examples

New section contains new examples.

14.1 New examples

New examples are nice.

14.1.1 Whyburn’s Curve

The well-known Jordan Curve Theorem says that a simple closed curve in the plane cuts the plane into two regions and is their common boundary. Thus, it is a cutting of the plane. The so called Warsaw Circle, obtained from the sin(1/x)-curve

S = {(0, y) ∈ ℝ^{2} : y ∈ [−1, 1]} ∪ {(x, sin(1/x)) ∈ ℝ^{2} : x ∈ (0, 1]}

by identifying the points (0, −1) and (1, sin 1) also is a cutting. But both these continua contain arcs, which certainly do not cut the plane. So, one can ask a question if there exists such a cutting X ⊂ ℝ^{2} that each nondegenerate subcontinuum of X also cuts ℝ^{2}. The question has been answered in the affirmative in 1930 by G. T. Whyburn who constructed in [3] a hereditary cutting of the plane, which is now called the Whyburn’s Curve. The present description of this peculiar continuum is a modification of the original one due to A. Lelek [1], to obtain some extra properties of the curve, and it follows Lelek’s sketch of the construction in [3, p. 117-119]. We say that a continuum X lying in the plane ℝ^{2} cuts the plane between points a and b if the points belong to different components of ℝ^{2} \ X. A continuum X is a cutting of ℝ^{2} if there are some two points such that X cuts ℝ^{2} between them. Take the sin(1/x)curve S and identify points (0, 1) and (0, −1) into one point r. The obtained curve (still considered as a subset of the plane) consists of a circle C and a sinusoidal curve approximating the circle, so it is a cutting of the plane. If we add to the curve an arc starting at r and having only this point in common with the curve, we get a continuum Y being also a cutting of the plane (namely it cuts the plane between a point a inside the circle C and a point b lying in ℝ^{2} \ Y outside C) and having the property that every subcontinuum of Y containing a point p of the sinusoidal part of Y and a point q lying on the added arc, also cuts the plane between a and b (see Figure A (I)-(II).).

The same curve Y can also be obtained as the intersection of a nested sequence of plane continua. Each continuum of the sequence has the form of a long strip with one loop and finitely many zig-zags. Consecutive strips are thinner and thinner, more and more narrow, each next being inscribed in the previous one in such a way that the next loop is contained in the previous loop which also contains one more zig-zag of the next strip (see Figure A (III)-(IV)). Consequently, the number of zig-zags in the next approximation is greater than that in the previous one, so these numbers tend to infinity. The common part of the all strips is just the curve Y, while the common part of the loops is the circle C contained in Y.

The Whyburn’s curve W can be obtained in a similar way, as the intersection of a nested sequence of strips in the plane, using the method called ”condensation of singularities”. Namely the singularity appearing in each neighborhood of C in Y is copied more and more times both in the loops as well as in the zig-zag parts of the consecutive strips. A little bit more precisely speaking, the strips are modified so that each next strip, magnifying the number of zig-zags in the loops and preserving previous loops, contains also new, smaller loops, distributed in a more and more dense manner inside the previous strip (see Figure A (V)) so that the diameters of the loops tend to zero as the number of steps of the construction tends to infinity. As a consequence of this property, for every two points p and q of the curve W one can find, in a sufficiently far approximating strip, a loop lying between these points. Roughly speaking, behavior of the curve W between p and q is such as that of the curve Y, i.e., each subcontinuum of W containing p and q cuts the plane between some points a and b determined by the mentioned loop.

The continuum W constructed is this way has, among others, the following properties (see [1]). 1) each subcontinuum of W is a cutting of the plane, and therefore W contains no arc; 2) each subcontinuum of W contains a homeomorphic copy of W; 3) W is hereditarily decomposable; and 4) W is a continuous image of the pseudo-arc ℙ. Since neither the pseudo-arc ℙ nor any subcontinuum of ℙ cuts the plane, this last property shows that a hereditary anti-cutting of ℝ^{2} can be continuously mapped onto a hereditary cutting.

[1] A. Lelek : On weakly chainable continua, Fund. Math. 51 (1962), 271– 282.

[2] A. Lelek : Zbiory, PZWS, Warszawa 1966.

[3] G. T. Whyburn : A continuum every subcontinuum of which separates the plane, Amer. J. Math. 52 (1930), 319–330.

14.1. NEW EXAMPLES 51 Figure 14.1.1: ( A ) construction of the Whyburn’s Curve

14.1.2 Pseudo-arc

A. Lelek, ZBIORY (Sets), PZWS Warszawa 1966, pages 90-95.The first heredi-

tarily indecomposable continuum has been constructed by B. Knaster in 1922. This continuum K, discovered by Knaster, is just named the pseudo-arc or Knaster’s continuum. The pseudo-arc K is obtained as the common part of a decreasing sequence of continua in the plane, every of which is the union of finitely many discs, forming a chain of sets. We will consider chains

D = (D_{1}, D_{2}, …, D_{k})

of discs D_{i} in the plane, having the property that any two consecutive disc D_{i} and D_{i+1} have interior points in common, i.e., that their intersection — nonempty by assumption — is not contained in the boundary of any of them (Figure A). Discs D_{i} are called links of the chain D. The union of all links of the chain D is a continuum. Observe that if we delete from the chain D a link which is not an end link, i.e., a link D_{i} such that 1 < i < k, then the union of the remaining links is a not connected set, having two components.

Let two chains of discs D and D′ be given. We say that the chain D′ refines the chain D if each link D′_{s} of the chain D′ is contained in the interior of at least one link D_{i} of the chain D, i.e., if for every D′_{s} ∈ D′ there exists a link D_{i} ∈ D such that D′_{s} ⊂ D_{i} and that the disc D′_{s} has no common points with the boundary of the disc D_{i} (Figure B).

We say that the chain D′ is crooked in the chain D if it refines D and if, for every pair of links D_{i} and D_{j} of the chain D such that

i + 2 < j,  (1)

and for every pair of links D′_{s} and D′_{v} of the chain D′, intersecting the links D_{i} and D_{j} respectively, i.e.,

D_{i} ∩ D′_{s} ≠ ∅ ≠ D_{j} ∩ D′_{v},

there are, in the chain D′, links D′_{t} and D′_{u}, lying between the link D′_{s} and D′_{v} in the same order, i.e.,

s < t < u < v or s > t > u > v,

such that

D′_{t} ⊂ D_{j−1} and D′_{u} ⊂ D_{i+1}.  (2)

In other words, a part of the chain D′ which is contained between the links D′_{s} and D′_{v} has to form a fold in the chain D between the links D_{i} and D_{j} (Figure C). The chain D′, to go from the link D_{i} to the link D_{j} has first to come to the link D_{j−1}, next go back to the link D_{i+1}, and only after this it can reach the link D_{j}. Such a fold must exist for every two links D_{i} and D_{j} having no adjoining (i.e. neighbor) link in common. Now, let there be given in the plane two points a and b, and an infinite sequence

D_{1}, D_{2}, …, D_{n}, …

of chains of discs, satisfying, for every n ∈ {1, 2, …}, the following conditions:

1^{o} the point a belongs to the first, and the point b to the last link of the chain D_{n};

2^{o} the diameter of every disc in the chain D_{n} is less than \frac{1}{n};

3^{o} the chain D_{n+1} is crooked in the chain D_{n}.

Thus, for example, the chain D_{2} is crooked in the chain D_{1}, and the chain D_{3} is crooked in the chain D_{2} (on Figure D only a part of the chain D_{3} is presented). The foldings are more and more condensed and they overlapped themselves. Denote by K_{n} the union of all links of the chain D_{n}. From condition

3^{o} it follows in particular that the each next chain refines the previous one, and therefore the continua K_{n} form a decreasing sequence

K_{1} ⊃ K_{2} ⊃ … ⊃ K_{n} ⊃ …

The Pseudo-arc K is defined as the common part of all continua K_{n}, i.e.,

K = ⋂{K_{n} : n ∈ ℕ}.

By virtue of condition 1^{o} points a and b belong to the continuum K, thus it is nondegenerate. We shall prove that the continuum K is hereditarily indecomposable.Let K′ ⊂ K be an arbitrary continuum. To prove that the continuum

K is indecomposable it is enough to show that every continuum Y ⊂ K′ distinct from K’ is nowhere dense in K′. Consider an arbitrary point p ∈ Y and number ε > 0.

Since K′ ≠ Y, hence there is a point q ∈ K′ \ Y which therefore is at a distance at least η > 0 from each point y ∈ Y, where η is a fixed number, independent from y. In fact, if not, then some points of the set would be arbitrarily close to the point q, and so the point q would belong to the closure of the set Y, which is impossible, because the set Y is closed, and it does not contain the point q. Let us take a natural number n so large that the inequalities are satisfied

\frac{2}{n} < ε and \frac{3}{n} < η.

Denote by D_{i} and D_{j} (with i ≤ j) the links of the chain D_{n} whose union contains the points p and q. Since p ∈ Y, hence ρ(p, q) ≥ η, (here ρ means the metric in the plane) and thus it follows that between the links D_{i} and D_{j} there are at least two links of the chain D_{n}. Indeed, in the opposite case the links D_{i} and D_{j} would have a neighbor common link, and choosing suitable points p′ and q′ in the intersections of this link with the links D_{i} and D_{j} we would have

ρ(p, q) ≤ ρ(p, p′) + ρ(p′, q′) + ρ(q′, q) < \frac{1}{n} + \frac{1}{n} + \frac{1}{n} = \frac{3}{n} < η

by condition 2^{o}. Thus we can assume that the indices i and j satisfy the inequality (1). Further, we can assume also that the point p belongs to one, and the point q to the other of the links D_{i} and D_{j}.

Denote by D′_{s} and D′_{v} the links of the chain D_{n+1} which contain points p and q, respectively. Additionally we assume that p ∈ D_{i}. In the opposite case, i.e. if p ∈ D_{j}, the further part of the proof runs in the same way (with changing of the roles of the links D_{i} and D_{j} as well as D_{i+1} and D_{j−1}).

So,

p ∈ D_{i} ∩ D′_{s} and q ∈ Dj ∩ D′_{v},

whence we infer, by virtue of condition 3^{o} and the definition of folding of chains, that there are two links D′_{t} and D′_{u} in the chain D_{n+1} which lie between the links D′_{s} and D′_{v} in the same order and which satisfy the inclusions (2). The union of the links of the chain D_{n+1} distinct from the link D′_{u} is a not connected set (see above) containing the point p in one component, and the point q in the other. Since the continuum K′ joins the points p and q and is contained in the union K_{n+1} of all links of the chain D_{n+1}, hence K′ must pass thru the link D′_{u}. Thus there is a point

x ∈ K′ ∩ D′_{u}

(Figure E). We shall prove that the point x does not belong to the continuum Y. Indeed, if it would be x ∈ Y, then by the same reason as previously for the continuum K′ the continuum Y would have to pass thru the link D′_{t}. Then there would exist a point

y ∈ Y ∩ D′_{t},

which would be in the link D_{j−1} according to (2). But the neighbor links D_{j} and D_{j−1} have their diameters less than \frac{1}{n} by condition 2^{o}, and therefore we would have

ρ(y, q) < \frac{2}{n} < \frac{3}{n} < η,

which is impossible because y ∈ Y.

So, the point x belongs to the set K′\Y and to the link D_{i+1} simultaneously. The neighbor links D_{i} and D_{i+1} have diameters less than \frac{1}{n}, whence it follows that

ρ(p, x) < \frac{2}{n} < ε.

Since ε was an arbitrary positive number, hence, in this way we have proved in that there are points of the set K′ \ Y which lie arbitrarily closely to the point p. The point p was an arbitrary point of the continuum Y. Thus we have proved that the continuum Y is nowhere dense in K′, and the proof of hereditary indecomposability of the continuum K is finished.

An example of a chain crooked in a chain with 7 links is on Figure F.

14.1.3 Dyadic Selenoid

The Dyadic Selenoid is obtained easily :-)

Figure 14.1.2: ( A ) a chain with links, construction of the Pseudo-arc

Figure 14.1.2: ( B ) refining a chain, construction of the Pseudo-arc

Figure 14.1.2: ( C ) crookedness used in the construction of the Pseudo-arc 56 CHAPTER 14. NEW EXAMPLES Figure 14.1.2: ( D ) crookedness used in the construction of the Pseudo-arc

Figure 14.1.2: ( E ) crookedness used in the construction of the Pseudo-arc 14.1. NEW EXAMPLES 57 Figure 14.1.2: ( F ) an example of a chain crooked in a chain with 7 liks, construction of the Pseudo-arc

Figure 14.1.3: ( A ) first step in the construction of the Dyadic Selenoid

14.1.4 Zig-zag Circle

The Zig-zag Circle is obtained easily :-)

14.1.5 M-Continuum

The M-Continuum is obtained easily :-)

14.1.6 Buckethandle

The Buckethandle is created from the Cantor Ternary Set C with this procedure:

(i) we join any two points a and b in C symmetric with respect to 1/2 with a semicircle in the upper half plane with the centre in (1/2, 0),

(ii) we join any two points of C in the interval 2/3^{n} ≤ x ≤ 3/3^{n}, n ≥ 1, with a semicircle with the centre in (5/(2 · 3^{n}), 0) in the lower half plane.

This continuum is often called the Knaster’s Buckethandle continuum (see Figure (A)).

This kind of curiosities in continuum theory is connected with common boundary of several plane regions. In 1904 A. Schönflies started to publish a sequence of papers [6] which became an important step in development of continuum theory. Relying heavily on intuition, he claimed that there do not exist three regions in the plane with common boundary. The claim was refused by L. E. J. Brouwer in 1910 [1] who constructed continua which are common boundary of three regions and showed that they are indecomposable, i.e., such 14.1. NEW EXAMPLES 59

Figure 14.1.4: ( A ) Zig-zag Circle Figure 14.1.5: ( A ) M-Continuum continua X that there are no nonempty proper subcontinua A and B of X with X = A∪B. The first example of such a continuum was given by Brouwer in 1910 (see Figure B, where the first several steps of his construction, simplified by Z. Janiszewski (1888–1920) [2, p. 114] are presented). Finally in 1922 B. Knaster (1893–1980) gave a nice description of this continuum (with the first full proof of its indecomposability) in [3, p. 209-210]. Now the continuum is referred to as the simplest indecomposable continuum, or the horse-shoe continuum, or the B-J-K continuum (for Brouwer, Janiszewski and Knaster).

Finally the common boundary problem of plane domains has been solved in 1928 by K. Kuratowski (1896–1980) who proved in [4] and [5] that every plane continuum which is the common boundary of n open domains either is indecomposable or is the union of two indecomposable continua whenever n ≥ 3, and when n = 2 it either is ”monostratic” or has a natural ”cyclic structure” in the sense that it is built up from layers naturally ordered in the same way as the individual points of the circle.

We can obtain a homeomorphic copy of the buckethandle continuum using the inverse limit

lim_{←}{X_{i}, f_{i}}_{i=1}^{∞}

where for each i = 1, 2, ⋯ let X_{i} = [0, 1] and f_{i}(t) = 2t for 0 ≤ t ≤ 1/2 and f_{i}(t) = −2t + 2 for 1/2 ≤ t ≤ 1.

We can find more in [65, p.22] and [55, p.205].

[1] L. E. J. Brouwer : Zur Analysis Situs, Math. Ann. 68(1910), 422–434.

[2] Z. Janiszewski: Sur les continus irréductibles entre deux points, Journal de l’Ecole Polytechnique (2) 16(1912), 79–170.

[3] K. Kuratowski: Théorie des continus irréductibles entre deux points I, Fund. Math. 3(1922), 200–231.

[4] K. Kuratowski: Sur les coupures irréductibles du plan, Fund. Math. 6(1924), 130–145.

[5] K. Kuratowski: Sur la structure des frontières communes à deux regions, Fund. Math. 12(1928), 20–42.

[6] A. Schönflies: Beirträge zur Theorie der Punktmengen I, Math. Ann. 58(1904), 195–244; II 59(1904), 129–160; III 62(1906), 286–236.

14.1.7 Cantor Interaction

Cantor Interaction is a Charatonic’s arcwise connected world ...

14.1.8 Pseudo-circle

Pseudo-circle is a strange continuum.

14.1.9 Semicircle Continuum

Semicircle Continuum is a nice continuum.

Figure 14.1.6: ( A ) Buckethandle

Let A = {(x, 0) ∈ ℝ^{2} : 0 ≤ x ≤ 1}; for each n = 1, 2, ⋯ and each k = 1, ⋯, 2^{n−1}, let

B_{n,k} = \{(x, y) ∈ ℝ^{2} : (x − \frac{2k − 1}{2^{n}})^{2} + y^{2} = 4^{−n}andy ≥ 0\};

for each n = 0, 1, ⋯ and each k = 1, ⋯, 3^{n}, let

C_{n,k} = \{(x, y) ∈ ℝ^{2} : (x − \frac{2k − 1}{2 · 3^{n}})^{2} + y^{2} = \frac{1}{4} · 9^{−n}andy ≤ 0\};

let

X = A ∪ [ ⋃_{n=1}^{∞} ( ⋃_{k=1}^{2^{n−1}} B_{n,k})] ∪ [ ⋃_{n=0}^{∞} ( ⋃_{k=1}^{3^{n}} C_{n,k})]

1) X is an example of an hlc continuum which is not regular.

2) X is the union of two regular continua. See Nadler 10.59, p.193.

14.1.10 Boxes Continuum

Boxes Continuum is a nice continuum.

Let A = {(x, 2^{−n+1}) ∈ ℝ^{2} : 0 ≤ x ≤ 1}; for each n = 0, 1, 2, ⋯ and each m = 0, ⋯, 2^{n+1}, let 62 CHAPTER 14. NEW EXAMPLES

Figure 14.1.6: ( B ) several steps in the construction of the Buckethandle 14.1. NEW EXAMPLES 63

Figure 14.1.7: ( A ) Cantor Interaction 64 CHAPTER 14. NEW EXAMPLES Figure 14.1.8: ( A ) A circular chain crooked in a circular chain with 6 links, construction of the Pseudo-circle 14.1. NEW EXAMPLES 65

Figure 14.1.9: ( A ) Semicircle Continuum Figure 14.1.10: ( A ) Union of two dendrites, a non-hlc Boxes Continuum

B_{n,m} = {(m · 2^{−n−1}, y) ∈ ℝ^{2} : 0 ≤ y ≤ 2^{−n}};

finally, let

X = [ ⋃_{n=1}^{∞} A_{n}] ∪ [ ⋃_{n=0}^{∞} ( ⋃_{m=0}^{2^{n+1}} B_{n,m})] .

1) X is an example of an non-hlc continuum which is not regular.

2) X is the union of two regular continua.

See Nadler 10.38, p.186.

14.2 New examples 2

New examples are nice.

14.2.1 Graph K_{3,3}

Graph K_{3,3} is a not embedable in ℝ^{2}.

14.2.2 Graph K_{5}

Graph K_{5} is a simple continuum.

14.2.3 Space Filling Curve

Space Filling Curve is a strange continuum.

14.2. NEW EXAMPLES 2 67 Figure 14.2.1: ( A ) Graph K_{3,3} Figure 14.2.2: ( A ) Graph K_{5} Figure 14.2.3: ( A ) An example of Space Filling Curve

14.2.4 Universal Arc-like Continuum

Universal Arc-like Continuum is a useful continuum.

14.2.5 Lakes of Wada

Lakes of Wada is a strange continuum.

14.2.6 Borsuk Fan

Borsuk Fan is a nice continuum.

14.2.7 Ingram’s Atriodic Continuum

INGRAM COMMENT (modified):

Figure A represents the first and second composite of a mapping f of a triod T = OA∪OB∪OC, onto itself. The mapping f has the property that f(O) = B, f(A) = C, f(B) = C, f(C) = C, f(C/2) = O, f(B/3) = O, f(B/2) = A/2, f(2B/3) = O, f(A/4) = O, f(A/2) = A, f(3A/4) = O and the mapping is piecewise linear. In the picture, the black square represents O, the green arm is the image of OA, the blue arm is the image of OB and the red arm is the image of OC. The inverse limit of the inverse sequence, T, f, is an atriodic tree-like continuum which is not chainable [Fund. Math., 77(1972), 99 - 107]. We call it an Ingram’s Atriodic Continuum.

If one imagines that the fat triod in the top of the Fgure A represents the first of a sequence of tree chains covering the inverse limit then the blue-green-red folded triod represents the next terms of the sequence of covers.

14.2.8 Double Warsaw Circle

We call it a Double Warsaw Circle.

14.2.9 Warsaw Circle

We call it a Warsaw Circle.

14.2.10 Crooked Warsaw Circle

We call it a Crooked Warsaw Circle.

14.3 New examples 3

New examples 3 are useful.

14.3. NEW EXAMPLES 3 69 Figure 14.2.4: ( A ) First steps in the construction of the Universal Arc-like Continuum 70 CHAPTER 14. NEW EXAMPLES

Figure 14.2.5: ( A ) A construction of the Lakes of Wada

Figure 14.2.6: ( A ) thick segments are ’above’ the thin ones, Borsuk Fan

14.3.1 Young Spiral

We call it a Young Spiral. See Roman Maňka : On spirals and fixed point property, Fund. Math. 144(1994), p. 1-9.

14.3.2 Young Space

We call it a Young Space. See Roman Maňka : On spirals and fixed point property, Fund. Math. 144(1994), p. 1-9.

14.3.3 Bing Space

We call it a Bing Space. See Roman Maňka : On spirals and fixed point property, Fund. Math. 144(1994), p. 1-9.

14.3.4 Universal Smooth Dendroid

We call it a Universal Smooth Dendroid.

14.3.5 Claytor’s Curve A

We call it a Claytor’s Curve A.

14.3.6 Claytor’s Curve B

We call it a Claytor’s Curve B.

72 CHAPTER 14. NEW EXAMPLES Figure 14.2.7: ( A ) First steps in the construction of the Ingram’s Atriodic Continuum 14.3. NEW EXAMPLES 3 73

Figure 14.2.8: ( A ) Double Warsaw Circle Figure 14.2.9: ( A ) Warsaw Circle Figure 14.2.10: ( A ) Crooked Warsaw Circle 74 CHAPTER 14. NEW EXAMPLES

Figure 14.3.1: ( A ) Young Spiral 14.3. NEW EXAMPLES 3 75

Figure 14.3.2: ( A ) Young Space 76 CHAPTER 14. NEW EXAMPLES

Figure 14.3.3: ( A ) Bing Space Figure 14.3.4: ( A ) Universal Smooth Dendroid 14.3. NEW EXAMPLES 3 77

Figure 14.3.5: ( A ) Claytor’s Curve A Figure 14.3.6: ( A ) Claytor’s Curve B Figure 14.3.7: ( A ) Sin Addiction Continuum

Figure 14.3.8: ( A ) Dumbbell

14.3.7 Sin Addiction Continuum

We call it a Sin Addiction Continuum.

14.3.8 Dumbbell

We call it a Dumbbell.

14.3.9 Figure Eight

We call it a Figure Eight.

Figure 14.3.9: ( A ) Figure Eight

Figure 14.3.10: ( A ) Noose

Figure 14.4.1: ( A ) Theta Curve

14.3.10 Noose

We call it a Noose.

14.4 New examples 4

New examples 4 are useful.

14.4.1 Theta Curve

We call it a Theta Curve.

14.4.2 Trapezoid Basins Continuum

We call it a Trapezoid Basins Continuum.

14.4.3 Dragon Continuum

We call it a Dragon Continuum. It is a fix point free continuum.

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

80 CHAPTER 14. NEW EXAMPLES Figure 14.4.2: ( A ) Trapezoid Basins Continuum 14.4. NEW EXAMPLES 4 81

Figure 14.4.3: ( A ) Dragon Continuum 82 CHAPTER 14. NEW EXAMPLES

Figure 14.4.3: ( B ) Dragon Continuum

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.[65, 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 [42, 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 [42, 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 [59, 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 [78, 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 [42, 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. [64, (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, [64, Theorem 15.2, p. 514], and it is continuous on a dense G_{δ} subset of C(X), [64, 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 [64, 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. [78, 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. [41, 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 [59, 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. [64, (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 [64, (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 [65], as well as in [55].

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. [65, 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 [4, 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 [59, 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 [59, 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 [59, 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

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 [71, 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 [59, 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

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 [55, §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 [8, 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. [42, 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. [64, Definition 16.10, p. 538]).

quasi-monotone

A surjective mapping f : X → Y between compact spaces is said to be (see [59, 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 [42, 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. [64, 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 [59, 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 [71, 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 [34, 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 [2, 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 [42, 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 [72, p. 146]).

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 [59, 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 [72, p. 152] and references therein).

weakly monotone

A surjective mapping f : X → Y between compact spaces is said to be (see [59, 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 [57, 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.

Appendix B

List of examples

n-dimensional hereditarily indecomposable continua (R.H.Bing), p. ?? Bing Space, p. 69 Borsuk Fan, p. 66 Boxes Continuum, p. 59 Buckethandle, p. 56 Cantor Comb, p. 41 Cantor Fan, p. 43 Cantor Interaction, p. 58 Cantor Meanders, p. 33 Cantor Snake, p. 33 Claytor’s Curve A, p. 69 Claytor’s Curve B, p. 69 Crooked Warsaw Circle, p. 66 Cruller, p. 34 Double Buckethandle, p. 34 Double Topologists’s Sin Curve, p. 32 Double Warsaw Circle, p. 66 Dragon Continuum, p. 77 Dumbbell, p. 76 Dyadic Selenoid, p. 52 Figure Eight, p. 76 Gehman Dendrite with Ski, p. 41 Gehman Dendrite, p. 40 Graph K_{5}, p. 64 Graph K_{3,3}, p. 64 Harmonic Fan, p. 42 Harmonic Shredded Fan, p. ?? Hilbert’s cube, p. ?? Ingram’s Atriodic Continuum, p. 66 Lakes of Wada, p. 66 Lelek Fan, p. 45 M-Continuum, p. 56 Makuchowski Twin, p. 42 Makuchowski Umbrella, p. 45 Menger’s universal curve. Its Anderson’s characterization., p. ?? Noose, p. 77 Open Sin, p. 31 Pseudo-arc, p. 51 Pseudo-circle, p. 58 Root, p. 36 Semicircle Continuum, p. 58 Sierpiński Tringle, p. 13 Sierpiński Universal Plane Curve, p. 9 Sin Addiction Continuum, p. 76 Snake, p. 32 Space Filling Curve, p. 64 The pseudoarc, p. ?? The pseudocircle, p. ?? Theta Curve, p. 77 Topologist’s Sin Curve, p. 31 Trapezoid Basins Continuum, p. 77 Universal Arc-like Continuum, p. 66 Universal Dendrite D_{3}, p. 37 Universal Dendrite D_{4}, p. 37 Universal Dendrite D_{6}, p. 40 Universal Smooth Dendroid, p. 69 Ważewski’s Universal Dendrite, p. 37 Warsaw Circle, p. 66 Whyburn’s Curve, p. 47 Young Space, p. 69 Young Spiral, p. 69 Zig-zag Circle, p. 56

Index

-connected, 84 -equivalent, 87 -chain, 83 -like, 91 -map, 91 -od, 98 -smooth, 86 -smooth at, 86

absolute end point, 81 absolute retract, 81 absolutely terminal continuum, 81 acyclic, 81 almost chainable, 81 aposyndetic, 82 aposyndetic at, 82 arc, 82 arc-smooth, 82 arc-smooth at a point, 82 arc-structure, 82 arms, 98 atomic, 83 atriodic, 83

biconnected, 83 bihomogeneous, 83 Bing Space, 69 Borel sets, 83 Borsuk Fan, 66 Boxes Continuum, 59 branch point, 83 Buckethandle, 56

Cantor 1/5-discontinuum, 34 Cantor Comb, 41 Cantor Fan, 43 Cantor Interaction, 58 Cantor Meanders, 33 Cantor Snake, 33 circle, 84 Claytor’s Curve A, 69 Claytor’s Curve B, 69 completely regular, 84 composant of, 84 composant of X belonging to, 84 confluent, 84 connected im kleinen, 85 continuous, 98 continuous selection of, 85 continuum, 85 contractible, 86 converge 0-regularly, 85 converges homeomorphically to the continuum, 85

convex, 85

Crooked Warsaw Circle, 66

Cruller, 34 curve, 86 cut point of, 86 cyclic element, 86

decomposable, 86

deformation retract of, 86

dendrite, 87 dendroid, 87 dispersion point, 87 Double Buckethandle, 34 Double Topologists’s Sin Curve, 32 Double Warsaw Circle, 66 Dragon Continuum, 77 Dumbbell, 76 Dyadic Selenoid, 52

equivalent, 87

monotonely, 87 explosion point, 87

Figure Eight, 76 finitely linear, 88 finitely Suslinean, 88 fixed point, 88 fixed set property, 88 free arc, 88

Gehman Dendrite, 40 Gehman Dendrite with Ski, 41 graph, 89 Graph K_{5}, 64 Graph K_{3,3}, 64

half-ray curve, 89 Harmonic Fan, 42 Hausdorff metric, 90 hereditarily, 89 hereditarily decomposable, 89 hereditarily equivalent, 89 hereditarily indecomposable, 89 hereditarily locally connected, 90 hereditarily unicoherent, 90 hereditary, 89 hlc, 90 homeomorphism

near, 93 homogeneous, 90

with respect to, 90 HU-terminal continuum, 90 hyperspace, 90

chainable, 84 chaotic, 84

strongly, 99

improper shore point, 99 indecomposable, 90 induced mapping, 91 Ingram’s Atriodic Continuum, 66 initial point, 82 irreducible, 91 irreducible about, 91 is of order less than equal to, 94 Knaster’s Buckethandle, 56

Lakes of Wada, 66 Lelek Fan, 45 light, 91 links, 83 local homeomorphism, 92 local separating point, 92 locally, 92 locally confluent, 92 locally convex at a point, 85

M-Continuum, 56 Makuchowski Twin, 42 Makuchowski Umbrella, 45 Menger Universal Curve, 10 minimal

openly, 93 MO-mapping, 92 monotone, 92 feebly, 87

mutually aposyndetic, 93

natural projection, 93

neat, 93

non-cut point of, 86

Noose, 77

OM-mapping, 93

open, 93 Open Sin, 31 orbit of, 94 order, 94 order preserving mapping, 94

point end, 87 ordinary, 95 periodic, 95 ramification, 96 recurent, 96 weak cut, 101

PR-property, 95

preserving retraction, 94

property

periodic-recurrent, 95 property of Kelley, 95 pseudo-arc, 95 Pseudo-arc, 51 Pseudo-circle, 58 pseudo-confluent, 95

quasi-monotone, 96

radially convex at, 96 rational, 96 real curve, 96 refinable, 96 regular, 97 Regular, 97 regular at, 97 retract, 97 retraction, 97 rigid, 97

strongly, 99 Root, 36

scattered, 97 selectible, 97 selection, 97 self-homeomorphic, 97

fixed ball, 88 point, 88

pointwise, 95 at a point, 95 strongly, 99

strongly, 99 semi-aposyndetic, 97 semi-confluent, 98 semi-continuous, 98 semi-continuum, 98 semi-locally connected, 98 Semicircle Continuum, 58 shore point, 99 Sierpiński Triangle, 13 Sierpiński Universal Plane Curve, 9 simple closed curve, 98 simple chain, 83 simple triod, 98 Sin Addiction Continuum, 76 smooth, 99 smooth at a point, 99 Snake, 32 Space Filling Curve, 64 strictly non-mutually aposyndetic, 93 strong deformation retract of, 87 strongly unicoherent, 99 strongly unicoherent2, 100 Suslinean, 100

terminal continuum, 100 Theta Curve, 77 thin, 100 top, 98 Topologist’s Sin Curve, 31 totally non-semi-locally connected,

98 translation, 100 Trapezoid Basins Continuum, 77 tree, 100 tree-like, 91 triod, 101

unicoherent, 101

at, 101 uniquely arcwise connected, 101 universal, 101

for, 101 Universal Arc-like Continuum, 66 Universal Dendrite D_{3}, 37 Universal Dendrite D_{4}, 37 Universal Dendrite D_{6}, 40 Universal Smooth Dendroid, 69

Warsaw Circle, 66 Ważewski’s Universal Dendrite, 37 weakly confluent, 101 weakly hereditarily unicoherent, 102 weakly chainable, 101 weakly monotone, 102 weakly smooth at a point, 102 weakly Suslinian, 102 Whyburn’s Curve, 47 widely connected, 102 width w(X), 102

Young Space, 69 Young Spiral, 69 110 INDEX Zig-zag Circle, 56

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