Property of Kelley for the Cartesian Products and Hyperspaces

A continuum X having the property of Kelley is constructed such that neither X × [0, 1], nor the hyperspace C(X), nor small Whitney levels in C(X) have the property of Kelley. This answers several questions asked in the literature

A Degree of Nonlocal Connectedness

To any continuum X weassign an ordinal number (or the symbol ∞) s(X), called the degree of nonlocal connectedness of X. We show that (1) the degree cannot be increased under continuous surjections; (2) for hereditarily unicoherent continua X, the degree of a subcontinuum of X is less than or equal to s(X); (3) s(C(X)) ≤ s(X), where C(X) denotes the hyperspace of subcontinua of a continuum X. We also investigate the degrees of Cartesian products and inverse limits. As an application weconstruct an uncountable family of metric continua X homeomorphic to C(X)

Generalized Homogeneity of Finite and of Countable Topological Spaces

Finite and countable topological spaces are investigated which are homogeneous, homogeneous with respect to open mappings or with respect to continuous ones. It is shown that for finite spaces all three concepts of homogeneity coincide, while for countable or for uncountable ones they are distinct. Some characterization of countable spaces that are homogeneous in either sense are found for the metric setting

On mapping properties and the property of Kelley

Mapping conditions are studied under which a continuum having the property of Kelley has this property hereditarily. The obtained results, related mainly to confluent mappings, extend some known assertions of the subject

Confluent Mappings and Arc Kelley Continua

A Kelley continuum X, also called a continuum with the property of Kelley, such that, for each p X, each subcontinuum K containing p is approximated by arc-wise connected continua containing p, is called an arc Kelley continuum. A continuum homeomorphic to the inverse limit of locally connected continua with confluent bonding maps is said to be confluently LC-representable. The main subject of the paper is a study of deep connections between the arc Kelley continua and confluent mappings. It is shown that if a continuum X admits, for each ε \u3e 0, a confluent ε-mapping onto a(n) (arc) Kelley continuum, then X itself is a(n) (arc) Kelley continuum. In particular each confluently LC-representable continuum is arc Kelley. It is also proved that if continua X and Y are confluently LC-representable, then also are their product X x Y and the hyperspaces 2^x and C(X)

Hereditarily Unicoherent Continua and Their Absolute Retracts

We investigate absolute retracts for classes of hereditarily unicoherent continua, tree-like continua, λ- dendroids, dendroids and some other related ones. The main results are: (1) the inverse limits of trees with confluent bonding mappings are absolute retracts of hereditarily unicoherent continua; (2) each tree-like continuum is embeddable in a special way in a tree-like absolute retract for the class of hereditarily unicoherent continua; (3) a dendroid is an absolute retract for hereditarily unicoherent continua if and only if it can be embedded as a retract into the Mohler-Nikiel universal smooth dendroid

Arc Approximation Property and Confluence of Induced Mappings

We say that a continuum X has the arc approximation property if every subcontinuum K of X is the limit of a sequence of arcwise connected subcontinua of X all containing a fixed point of K. This property is applied to exhibit a class of continua Y such that confluence of a mapping f : X - Y implies confluence of the induced mappings 2^f : 2^x - @^y and C(f) : C(x) - C(y). The converse implications are studied and similar interrelations are considered for some other classes of mappings, related to confluent ones

More on unicoherence at subcontinua

Studies are continued of unicoherence of a continuum X at its subcontinuum Y. Relations are analyzed between unicoherence of X at Y, unicoherence of either X or Y, and structure of components of the complement X Y. The obtained results generalize certain teorems proved in cite [OP]. Further, it is shown that terminality of Y implies unicoherence of X at Y. Applications are shown of this result to compactifications of a ray

Ri-continua and hyperspaces

AbstractIt is proved that if a continuum X contains an Ri-continuum for some iϵ{1,2,3}, then the hyperspaces 2x and C(X) contain Ri-continua, therefore they are not contractible. Moreover, 2x has no confluent Whitney map. Some examples concerning this subject are given and som