Čech homology for movable compacta
Cells and cubes in hyperspaces
Centers of a dendroid
A bottleneck in a dendroid is a continuum that intersects every arc connecting two non-empty open sets. Piotr Minc proved that every dendroid contains a point, which we call a center, contained in arbitrarily small bottlenecks. We study the effect that the set of centers in a dendroid has on its structure. We find that the set of centers is arc connected, that a dendroid with only one center has uncountably many arc components in the complement of the center, and that, in this case, every open set...
Chainable continua and homeomorphisms of the plane onto itself
Characterizing chainable, tree-like, and circle-like continua
We prove that a continuum X is tree-like (resp. circle-like, chainable) if and only if for each open cover 𝓤₄ = {U₁,U₂,U₃,U₄} of X there is a 𝓤₄-map f: X → Y onto a tree (resp. onto the circle, onto the interval). A continuum X is an acyclic curve if and only if for each open cover 𝓤₃ = {U₁,U₂,U₃} of X there is a 𝓤₃-map f: X → Y onto a tree (or the interval [0,1]).
Characterizing the arc by composition of functions
Characterizing the interval and circle by composition of functions
Circularity of graphs and continua: topology
Clumps of continua
Compactification of pointed 1-movable spaces
Compactifying the space of homeomorphisms
Complexity of curves
We show that each of the classes of hereditarily locally connected, finitely Suslinian, and Suslinian continua is Π₁¹-complete, while the class of regular continua is Π₀⁴-complete.
Composant-like decompositions
The body of this paper falls into two independent sections. The first deals with the existence of cross-sections in -decompositions. The second deals with the extensions of the results on accessibility in the plane.
Composants of the horseshoe
The horseshoe or bucket handle continuum, defined as the inverse limit of the tent map, is one of the standard examples in continua theory as well as in dynamical systems. It is not arcwise connected. Its arcwise components coincide with composants, and with unstable manifolds in the dynamical setting. Knaster asked whether these composants are all homeomorphic, with the obvious exception of the zero composant. Partial results were obtained by Bellamy (1979), Dębski and Tymchatyn (1987), and Aarts...
Compositions of confluent mappings and some other classes of functions
Concerning atriodic tree-like continua
Concerning closed quasi-orders on hereditarily unicoherent continua
Concerning decompositions of continua
Concerning hereditarily locally connected continua
Concerning indecomposable continua and upper semicontinuous collections of nondegenerate continua