\contentsline {chapter}{\numberline {0} PRELIMINARIES}{7}
\contentsline {chapter}{\numberline {1} LOCALLY CONNECTED CONTINUA}{9}
\contentsline {section}{\numberline {1.1} Basic examples: an arc, a circle, $n$-cube $\@mathbb I^n$, Hilbert cube}{9}
\contentsline {section}{\numberline {1.2} Graphs}{9}
\contentsline {section}{\numberline {1.3} Dendrites}{9}
\contentsline {subsection}{\numberline {1.3.1} {Preliminary properties} }{10}
\contentsline {subsection}{\numberline {1.3.2} {Simple examples} }{11}
\contentsline {subsection}{\numberline {1.3.3} {The locally connected fan} }{11}
\contentsline {subsection}{\numberline {1.3.4} {The locally connected combs} }{11}
\contentsline {subsection}{\numberline {1.3.5} {Universal dendrites} }{12}
\contentsline {subsection}{\numberline {1.3.6} {Wa\.zewski universal dendrite} }{12}
\contentsline {subsection}{\numberline {1.3.7} {Universal dendrites of order $m$} }{13}
\contentsline {subsection}{\numberline {1.3.8} {Other universal dendrites} }{17}
\contentsline {subsection}{\numberline {1.3.9} {The dendrite $P$} }{18}
\contentsline {subsection}{\numberline {1.3.10} {Self-homeomorphic dendrites} }{18}
\contentsline {subsection}{\numberline {1.3.11} {Monotone equivalence and monotone homogeneity} }{19}
\contentsline {subsection}{\numberline {1.3.12} {Omiljanowski dendrite} }{19}
\contentsline {subsection}{\numberline {1.3.13} {Chaotic and rigid dendrites} }{23}
\contentsline {subsection}{\numberline {1.3.14} {Miller dendrite} }{24}
\contentsline {subsection}{\numberline {1.3.15} {Dendrites of de Groot-Wille type} }{24}
\contentsline {subsection}{\numberline {1.3.16} {Modified Miller dendrites} }{26}
\contentsline {subsection}{\numberline {1.3.17} {Dendrites with the closed set of end points} }{26}
\contentsline {subsection}{\numberline {1.3.18} {Gehman dendrite} }{27}
\contentsline {subsection}{\numberline {1.3.19} {Modifications of the Gehman dendrite} }{27}
\contentsline {subsection}{\numberline {1.3.20} {Mapping hierarchy of dendrites} }{28}
\contentsline {section}{\numberline {1.4} Cyclic examples of locally connected continua}{29}
\contentsline {subsection}{\numberline {1.4.1} {Menger universal continua} }{29}
\contentsline {subsection}{\numberline {1.4.2} {Sierpi\'nski carpet} }{31}
\contentsline {subsection}{\numberline {1.4.3} {Menger universal curve} }{34}
\contentsline {section}{\numberline {1.5} 2-dimensional locally connected continua }{36}
\contentsline {section}{\numberline {1.6} Higher dimensional locally connected continua }{36}
\contentsline {section}{\numberline {1.7} Variations of local connectedness}{36}
\contentsline {chapter}{\numberline {2} JOINING PROPERTIES}{37}
\contentsline {section}{\numberline {2.1} Cut points and separating points (also local)}{37}
\contentsline {section}{\numberline {2.2} Arcwise connectedness; arc components}{37}
\contentsline {section}{\numberline {2.3} Continuum chainability}{37}
\contentsline {section}{\numberline {2.4} Various kinds of connectedness ($\lambda $-connectedness, $\delta $-connectedness, etc.)}{37}
\contentsline {chapter}{\numberline {3} DECOMPOSABLE AND INDECOMPOSABLE CONTINUA}{39}
\contentsline {section}{\numberline {3.1} Decomposable continua (also hereditary)}{39}
\contentsline {section}{\numberline {3.2} Indecomposable continua (also hereditary)}{39}
\contentsline {subsection}{\numberline {3.2.1} {Knaster continua} }{39}
\contentsline {section}{\numberline {3.3} Unicoherent continua (also hereditary)}{43}
\contentsline {section}{\numberline {3.4} Discoherent continua}{43}
\contentsline {section}{\numberline {3.5} Multicoherent continua}{43}
\contentsline {chapter}{\numberline {4} IRREDUCIBLE CONTINUA}{45}
\contentsline {section}{\numberline {4.1} Elementary examples}{45}
\contentsline {subsection}{\numberline {4.1.1} {Sin curve} }{45}
\contentsline {subsection}{\numberline {4.1.2} {Cantor organ and accordion} }{47}
\contentsline {section}{\numberline {4.2} Arc-like (chainable) continua}{48}
\contentsline {section}{\numberline {4.3} Irreducible circle-like continua}{48}
\contentsline {subsection}{\numberline {4.3.1} {Solenoids} }{48}
\contentsline {section}{\numberline {4.4} Compactifications of the real line and the real half-line}{51}
\contentsline {section}{\numberline {4.5} Other examples}{51}
\contentsline {chapter}{\numberline {5} USC DECOMPOSITIONS OF CONTINUA}{53}
\contentsline {section}{\numberline {5.1} Decomposition spaces}{53}
\contentsline {section}{\numberline {5.2} Upper semi-continuous decompositions}{53}
\contentsline {section}{\numberline {5.3} Continuous decompositions}{53}
\contentsline {chapter}{\numberline {6} CURVES}{55}
\contentsline {section}{\numberline {6.1} Order of a space at a point}{55}
\contentsline {section}{\numberline {6.2} Case-Chamberlin examples}{55}
\contentsline {section}{\numberline {6.3} Curves without arcs}{55}
\contentsline {section}{\numberline {6.4} Covering defined classes (inverse limit expansions)}{55}
\contentsline {section}{\numberline {6.5} Other examples}{55}
\contentsline {chapter}{\numberline {7} FAMILIES OF CONTNINUA}{57}
\contentsline {section}{\numberline {7.1} Universal elements}{57}
\contentsline {section}{\numberline {7.2} Common models}{57}
\contentsline {section}{\numberline {7.3} Families of incomparable continua}{57}
\contentsline {chapter}{\numberline {8} PLANE CONTINUA}{59}
\contentsline {section}{\numberline {8.1} Embeddability of continua in surfaces}{59}
\contentsline {section}{\numberline {8.2} Planability in general}{59}
\contentsline {section}{\numberline {8.3} Planability of graphs on of locally connected continua}{59}
\contentsline {section}{\numberline {8.4} Separation of the plane}{59}
\contentsline {chapter}{\numberline {9} MAPPINGS}{61}
\contentsline {section}{\numberline {9.1} Various kinds of mappings and relations between them}{61}
\contentsline {chapter}{\numberline {10} CONTINUA WITH SPECIAL MAPPING PROPERTIES}{63}
\contentsline {section}{\numberline {10.1} Existence of special mappings}{63}
\contentsline {section}{\numberline {10.2} Self-similar continua}{63}
\contentsline {section}{\numberline {10.3} Self-homeomorphic continua}{63}
\contentsline {section}{\numberline {10.4} Rigid continua}{63}
\contentsline {section}{\numberline {10.5} Chaotic continua}{64}
\contentsline {section}{\numberline {10.6} Absolute retracts; spaces $LC^n$ and $LC^{\infty }$}{64}
\contentsline {section}{\numberline {10.7} Absolute retracts for various classes of continua}{64}
\contentsline {section}{\numberline {10.8} Class (W) and other classes defined by mappings}{64}
\contentsline {section}{\numberline {10.9} Homogeneity}{64}
\contentsline {section}{\numberline {10.10} Variations of homogeneity }{64}
\contentsline {section}{\numberline {10.11} Generalized homogeneity}{64}
\contentsline {section}{\numberline {10.12} Embeddability in $\@mathbb R^n$}{64}
\contentsline {section}{\numberline {10.13} Weakly chainable continua}{64}
\contentsline {section}{\numberline {10.14} Fixed point property}{64}
\contentsline {chapter}{\numberline {11} CONTINUA AS FRACTALS}{65}
\contentsline {chapter}{\numberline {12} DYNAMICAL SYSTEMS}{67}
\contentsline {section}{\numberline {12.1} Attractors}{67}
\contentsline {chapter}{\numberline {13} OTHER PROPERTIES}{69}
\contentsline {section}{\numberline {13.1} Colocal connectedness}{69}
\contentsline {section}{\numberline {13.2} Aposyndesis and its variations}{69}
\contentsline {section}{\numberline {13.3} Property of Kelley (also hereditary)}{69}
\contentsline {section}{\numberline {13.4} Smoothness}{69}
\contentsline {section}{\numberline {13.5} Arc-smoothness}{69}
\contentsline {section}{\numberline {13.6} $C^*$-smoothness}{69}
\contentsline {chapter}{\numberline {14} NEW FIGURES}{71}
\contentsline {section}{\numberline {14.1} New examples 1}{71}
\contentsline {subsection}{\numberline {14.1.1} {Whyburn's curve $W$} }{71}
\contentsline {subsection}{\numberline {14.1.2} {Pseudo-arc} }{71}
\contentsline {subsection}{\numberline {14.1.3} {Zig-zag circle} }{71}
\contentsline {subsection}{\numberline {14.1.4} {M-continuum} }{71}
\contentsline {subsection}{\numberline {14.1.5} {Cantor interaction} }{71}
\contentsline {subsection}{\numberline {14.1.6} {Pseudo-circle} }{71}
\contentsline {subsection}{\numberline {14.1.7} {Semicircle continuum} }{71}
\contentsline {subsection}{\numberline {14.1.8} {Boxes continuum} }{80}
\contentsline {section}{\numberline {14.2} New examples 2}{80}
\contentsline {subsection}{\numberline {14.2.1} {Graph $K_{3,3}$} }{80}
\contentsline {subsection}{\numberline {14.2.2} {Graph $K_5$} }{80}
\contentsline {subsection}{\numberline {14.2.3} {Space filling curve} }{81}
\contentsline {subsection}{\numberline {14.2.4} {Universal arc-like continuum} }{81}
\contentsline {subsection}{\numberline {14.2.5} {Lakes of Vada} }{81}
\contentsline {subsection}{\numberline {14.2.6} {Borsuk fan} }{81}
\contentsline {subsection}{\numberline {14.2.7} {Ingram's atriodic continuum} }{84}
\contentsline {subsection}{\numberline {14.2.8} {Double Warsaw circle} }{84}
\contentsline {subsection}{\numberline {14.2.9} {Warsaw circle} }{84}
\contentsline {subsection}{\numberline {14.2.10} {Crooked Warsaw circle} }{84}
\contentsline {section}{\numberline {14.3} New examples 3}{84}
\contentsline {subsection}{\numberline {14.3.1} {Young spiral} }{84}
\contentsline {subsection}{\numberline {14.3.2} {Young space} }{84}
\contentsline {subsection}{\numberline {14.3.3} {Bing space} }{89}
\contentsline {subsection}{\numberline {14.3.4} {Universal smooth dendroid} }{89}
\contentsline {subsection}{\numberline {14.3.5} {Claytor's curve A} }{89}
\contentsline {subsection}{\numberline {14.3.6} {Claytor's curve B} }{89}
\contentsline {subsection}{\numberline {14.3.7} {Sin addiction continuum} }{89}
\contentsline {subsection}{\numberline {14.3.8} {Dumbbell} }{92}
\contentsline {subsection}{\numberline {14.3.9} {Figure Eight} }{92}
\contentsline {subsection}{\numberline {14.3.10} {Noose} }{92}
\contentsline {section}{\numberline {14.4} New examples 4}{93}
\contentsline {subsection}{\numberline {14.4.1} {Theta curve} }{93}
\contentsline {subsection}{\numberline {14.4.2} {Trapezoid basins continuum} }{93}
\contentsline {subsection}{\numberline {14.4.3} {Dragon continuum} }{93}
\contentsline {section}{\numberline {14.5} New examples 5}{96}
\contentsline {subsection}{\numberline {14.5.1} {Harmonic fan} }{96}
\contentsline {subsection}{\numberline {14.5.2} {Harmonic shredded fan} }{96}
\contentsline {subsection}{\numberline {14.5.3} {Cantor fan} }{96}
\contentsline {subsection}{\numberline {14.5.4} {Lelek fan} }{96}
\contentsline {section}{\numberline {14.6} New examples 6}{98}
\contentsline {subsection}{\numberline {14.6.1} {Sierpi\'nski triangle} }{98}
\contentsline {subsection}{\numberline {14.6.2} {Makuchowski umbrella} }{98}
\contentsline {subsection}{\numberline {14.6.3} {Snake} }{98}
\contentsline {subsection}{\numberline {14.6.4} {Cantor snake} }{98}
\contentsline {subsection}{\numberline {14.6.5} {Cruller} }{98}
\contentsline {section}{\numberline {14.7} New examples 7}{98}
\contentsline {subsection}{\numberline {14.7.1} {Cantor comb} }{98}
\contentsline {subsection}{\numberline {14.7.2} {Gehman dendrite on ski} }{98}
\contentsline {subsection}{\numberline {14.7.3} {Makuchowski twin} }{102}
\contentsline {subsection}{\numberline {14.7.4} {Cantor function comb} }{102}
\contentsline {subsection}{\numberline {14.7.5} {Oversteegen's dendroid} }{102}
\contentsline {chapter}{\numberline {A}Definitions}{107}
\contentsline {chapter}{\numberline {B}List of examples}{131}