Contents

• Contents
• CONNECTEDNESS
  • Connectedness, arcwise connectedness, continuumwise connectedness (continuum chainability)
  • A cut point (the complement is not connected); a separating point (the complement is not continuumwise connected)
  • Components, quasi-components, composants. Arc-components.
• LOCAL CONNECTEDNESS
  • Local connectedness and related concepts (connectedness im kleinen, aposyndesis, semi-local connectedness); relations between them.
• DIMENSION
  • The concept of a curve. Mapping of [0,1] onto [0,1]².
  • Menger-Urysohn dimension of a space at a point.
  • Universal spaces (in general).
    • Hilbert's cube
    • Sierpiski Universal Plane Curve
    • Menger's universal curve. Its Anderson's characterization.
  • Universal Menger's compacta. Their Bestvina's characterization.
  • Nöbeling's examples. (?).
  • 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.
• ORDER OF A POINT
  • Menger-Urysohn's order of a point.
  • Curves of a finite order.
  • Urysohn's examples: for each positive integer there is a continuum X(n) consisting solely of points of order n and 2n - 2.
    • Sierpiski Tringle
  • Urysohn's examples: for there is a continuum consisting solely of points of order
• DENDRITES
  • Dendrites and their basic properties.
  • Universal dendrites of order . Various constructions.
  • Waewski's universal dendrite.
  • Local dendrites.
• INDECOMPOSABILITY
  • Early examples.
  • Common boundary of three or more domains. Examples by R. L. Wilder and M. Lubaski.
  • Indecomposable continua via inverse limits.
• IRREDUCIBLE CONTINUA; DECOMPOSITIONS
  • ???
• SPECIAL CONTINUA
  • Hereditarily decomposable curves without arcs (Z. Janiszewski).
  • Whyburn's hereditary cut of the plane.
  • Hereditarily indecomposable continua.
    • The pseudoarc
    • The pseudocircle
    • n-dimensional hereditarily indecomposable continua (R.H.Bing)
  • The circle of pseudoarcs.
  • -like continua.
• UNIVERSAL SPACES
  • Universal arc-like continuum(Schori).
  • Universal circle-like continuum(Rogers).
  • Some universal curves (Krasinkiewicz and Minc).
  • Mohler-Nikiel universal smooth dendroid.
• MAPPINGS
  • Mapping properties -- families of continua.
  • Various kinds of mappings.
  • Fixed point theory.
• HOMOGENEOUS SPACES
  • Homogeneity.
  • Homogeneous continua. Solenoids.
  • Homogeneity of Menger compacta (Anderson, W. Lewis).
  • Variants of homogeneity; Knaster problem; K. Kuperberg's results.
  • Generalized homogeneity.
• HYPERSPACES
  • Hyperspaces are terrible ;-)
• Lost continua
  • Arc-like continua
    • Topologist's Sin Curve
    • Open Sin
    • Double Topologists's Sin Curve
    • Snake
    • Cantor Meanders
    • Cantor Snake
    • Cruller
    • Double Buckethandle
  • Self-homeomorphic continua
    • Root
  • Dendrites
    • Waewski's Universal Dendrite
    • Universal Dendrite D₃
    • Universal Dendrite D₄
    • Universal Dendrite D₆
    • Gehman Dendrite
  • Dendroids
    • Cantor Comb
    • Gehman Dendrite with Ski
    • Makuchowski Twin
  • Fans
    • Harmonic Fan
    • Harmonic Shredded Fan
    • Cantor Fan
    • Lelek Fan
  • Planable continua
    • Makuchowski Umbrella
• New examples
  • New examples
    • Whyburn's Curve
    • Pseudo-arc
    • Dyadic Selenoid
    • Zig-zag Circle
    • M-Continuum
    • Buckethandle
    • Cantor Interaction
    • Pseudo-circle
    • Semicircle Continuum
    • Boxes Continuum
  • New examples 2
    • Graph K_3,3
    • Graph K₅
    • Space Filling Curve
    • Universal Arc-like Continuum
    • Lakes of Wada
    • Borsuk Fan
    • Ingram's Atriodic Continuum
    • Double Warsaw Circle
    • Warsaw Circle
    • Crooked Warsaw Circle
  • New examples 3
    • Young Spiral
    • Young Space
    • Bing Space
    • Universal Smooth Dendroid
    • Claytor's Curve A
    • Claytor's Curve B
    • Sin Addiction Continuum
    • Dumbbell
    • Figure Eight
    • Noose
  • New examples 4
    • Theta Curve
    • Trapezoid Basins Continuum
    • Dragon Continuum
• Definitions
• List of examples
• Index
• Bibliography