Compact-like properties in hyperspaces

Abstract

$\mathcal{CL}(X)$ and $\mathcal{K}(X)$ denote thehyperspaces of non-empty closed and non-empty compact subsets of$X$, respectively, with the Vietoris topology. For an infinitecardinal number $\alpha$, a space $X$ is $\alpha$-hyperbounded iffor every family $\{S_{\xi}:\xi<\alpha\}$ of non-empty compactsubsets of $X$, $Cl_X(\bigcup_{\xi<\alpha}S_{\xi})$ is a compact set,and a space $X$ is pseudo-$\omega$-bounded if for each countablefamily $\mathcal{U}$ of non-empty open subsets of $X$, there exists acompact set $K\subseteq X$ such that each element in $\mathcal{U}$ hasa non-empty intersection with $K$. We prove that $X$ is$\alpha$-hyperbounded if and only if $\mathcal{K}(X)$ is$\alpha$-hyperbounded, if and only if $\mathcal{K}(X)$ is initially$\alpha$-compact. Moreover, $\mathcal{K}(X)$ is pseudocompact if andonly if $X$ is pseudo-$\omega$-bounded. Also, we show than if$\mathcal{K}(X)$ is normal and $C^{*}$-embbeded in $\mathcal{CL}(X)$, then$X$ is $\omega$-hyperbounded, and $X$ is $\alpha$-bounded if andonly if $X$ is $\alpha$-hyperbounded, for every infinite cardinalnumber $\alpha$.