Given a discrete group $G$, we consider the set $ℒ\left(G\right)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $ℒ\left(G\right)$ into the Cantor cube ${\{0,1\}}^G$. We show that the cellularity $c\left(ℒ\left(G\right)\right)\le {\aleph }_0$ for every abelian group $G$, and, for every infinite cardinal $\tau$, we construct a group $G$ with $c\left(ℒ\right(G\left)\right)=\tau$.

A homeomorphism f : X → X of a compactum X is expansive (resp. continuum-wise expansive) if there is c > 0 such that if x, y ∈ X and x ≠ y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ ℤ such that $d(f^n\left(x\right),f^n\left(y\right))>c$ (resp. $diam{f}^n\left(A\right)>c$). We prove the following theorem: If f is a continuum-wise expansive homeomorphism of a compactum X and the covering dimension of X is positive (dim X > 0), then there exists a σ-chaotic continuum Z = Z(σ) of f (σ = s or σ = u), i.e. Z is a nondegenerate subcontinuum...

It is shown that the hyperspace $Cl{d}_H\left(X\right)$ (resp. $Bd{d}_H\left(X\right)$) of non-empty closed (resp. closed and bounded) subsets of a metric space (X,d) is homeomorphic to ℓ₂ if and only if the completion X̅ of X is connected and locally connected, X is topologically complete and nowhere locally compact, and each subset (resp. each bounded subset) of X is totally bounded.

A net in a Hausdorff uniform space is called cofinally Cauchy if for each entourage, there exists a cofinal (rather than residual) set of indices whose corresponding terms are pairwise within the entourage. In a metric space equipped with the associated metric uniformity, if each cofinally Cauchy sequence has a cluster point, then so does each cofinally Cauchy net, and the space is called cofinally complete. Here we give necessary and sufficient conditions for the nonempty closed subsets of the...

Let (X, d) be a metric space and CL(X) the family of all nonempty closed subsets of X. We provide a new proof of the fact that the coincidence of the Vietoris and Wijsman topologies induced by the metric d forces X to be a compact space. In the literature only a more involved and indirect proof using the proximal topology is known. Here we do not need this intermediate step. Moreover we prove that (X, d) is boundedly compact if and only if the bounded Vietoris and Wijsman topologies on CL(X) coincide....

It is shown that every metrizable consonant space is a Cantor set-selector. Some applications are derived from this fact, also the relationship is discussed in the framework of hyperspaces and Prohorov spaces.

Let $X$ be a metric continuum. Let $F_n\left(X\right)$ denote the hyperspace of nonempty subsets of $X$ with at most $n$ elements. We say that the continuum $X$ has unique hyperspace $F_n\left(X\right)$ provided that the following implication holds: if $Y$ is a continuum and $F_n\left(X\right)$ is homeomorphic to $F_n\left(Y\right)$, then $X$ is homeomorphic to $Y$. In this paper we prove the following results: (1) if $X$ is an indecomposable continuum such that each nondegenerate proper subcontinuum of $X$ is an arc, then $X$ has unique hyperspace $F_2\left(X\right)$, and (2) let $X$ be an arcwise connected...

For a space $Z$, we denote by $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\le 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable if and...