On confluently graph-like compacta

Colloquium Mathematicum, 2003

We investigate absolute retracts for hereditarily unicoherent continua, and also the continua that have the arc property of Kelley (i.e., the continua that satisfy both the property of Kelley and the arc approximation property). Among other results we prove that each absolute retract for hereditarily unicoherent continua (for tree-like continua, for A-dendroids, for dendroids) has the arc property of Kelley. 1. Introduction. It is well known that absolute retracts (for the class of all compacta) are locally connected. This is also the case of absolute retracts for many smaller but important classes of spaces (see and). Following MaCkowiak's ideas from , in the present paper we study the class AR(1-fU) of absolute retracts for the class 1-fU of hereditarily unicoherent continua (and also absolute retracts for tree-like continua, A-dendroids, dendroids). These classes appear in a natural way in various regions of mathematical interest and are among the most extensively studied classes of continua. Therefore their absolute retracts seem to be worth a special attention. According to MaCkowiak's result (see Corollaries 4 and 5, pp. 181 and 183]), which was also independently proved by David P. Bellamy (unpublished; see [25, "added in proof", p. 183]), the bucket handle continuum and the Cantor fan belong to AR(1-fU). Thus the members of AR(1-fU) need not be locally connected. Recently, the authors proved [5] that every inverse limit of trees with confluent bonding mappings belongs to AR(1-fU). Consequently, we have a new large class of (not necessarily locally connected) continua that belong to AR(1-fU). Other authors' results concerning the class AR(1-fU) are presented in [5], [6], [7] and [11].

Topology and its Applications, 2007

We construct a path-connected homogenous compactum with cellularity c that is not homeomorphic to any product of dyadic compacta and first countable compacta. We also prove some closure properties for classes of spaces defined by various connectifiability conditions. One application is that every infinite product of infinite topological sums of T i spaces has a T i pathwise connectification, where i ∈ {1, 2, 3, 3 1 2 }.

Proceedings of the American Mathematical Society, 1987

In 1976 Eberhart, Fugate, and Gordh proved that the weakly confluent image of a graph is a graph. A much weaker condition on the map is introduced called partial confluence, and it is shown that the image of a graph is a graph if and only if the map is partially confluent. In addition, it is shown that certain properties of one-dimensional continua are preserved by partially confluent maps, generalizing theorems of Cook and Lelek, Tymchatyn and Lelek, and Grace and Vought. Also, some continua in addition to graphs are shown to be the images of partially confluent maps only.

The Journal of Symbolic Logic, 2020

We prove that the homeomorphism relation between compact spaces can be continuously reduced to the homeomorphism equivalence relation between absolute retracts, which strengthens and simplifies recent results of Chang and Gao, and Cieśla. It follows then that the homeomorphism relation of absolute retracts is Borel bireducible with the universal orbit equivalence relation. We also prove that the homeomorphism relation between regular continua is classifiable by countable structures and hence it is Borel bireducible with the universal orbit equivalence relation of the permutation group on a countable set. On the other hand we prove that the homeomorphism relation between rim-finite metrizable compacta is not classifiable by countable structures.

Commentationes Mathematicae Universitatis Carolinae, 2019

We call a function f : X → Y P-preserving if, for every subspace A ⊂ X with property P , its image f (A) also has property P. Of course, all continuous maps are both compactness-and connectedness-preserving and the natural question about when the converse of this holds, i.e. under what conditions is such a map continuous, has a long history. Our main result is that any non-trivial product function, i.e. one having at least two non-constant factors, that has connected domain, T 1 range, and is connectednesspreserving must actually be continuous. The analogous statement badly fails if we replace in it the occurrences of "connected" by "compact". We also present, however, several interesting results and examples concerning maps that are compactness-preserving and/or continuum-preserving. This paper is dedicated to the memory of Bohuslav Balcar.

Colloquium Mathematicum, 2011

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]).

Journal of the London Mathematical Society, 1986

The purpose of this paper is to prove several results concerning the shape category of a metric compactum in the sense of K. Borsuk. Among other things, we prove that if there exists a refinable map from a compactum X onto a compactum Y then the shape categories of A' and Y are coincident. We also characterize the shape category in terms of deformations of the diagonal. Finally, we introduce a new numerical shape invariant under the name of coefficient of movability and give a relation between the coefficient of movability of the total space, the coefficient of movability of the fibre and the shape category of the base space of a shape fibration.

Pacific Journal of Mathematics, 1973

arXiv (Cornell University), 2010

Given a class T of tree-like continua (and a cardinal number n) we define a compact Hausdorff space X to be T-like (resp. n-T-like) if for each open cover U of X (of cardinality |U|<n+1) the space X admits a U-map onto a spaces from the class T. Our principal result says that a compact Hausdorff space X is T-like if and only if it is 4-T-like. In particular, X is chainable if and only if it is 4-chainable. For 3-T-like spaces we have another characterization: a compact Hausdorff space X is 3-T-like if and only if X is an acyclic curve.

Bulletin of The Australian Mathematical Society, 2003

A compactum K ⊂ R 2 is said to be basically embed-