Abstract

$\mathcal{CL}(X)$ and $\mathcal{K}(X)$ denote the hyperspaces of non-empty closed and non-empty compact subsets of $X$, respectively, with the Vietoris topology. For an infinite cardinal number $\alpha$, a space $X$ is $\alpha$-hyperbounded if for every family $\{S_{\xi}:\xi<\alpha\}$ of non-empty compact subsets of $X$, $Cl_X(\bigcup_{\xi<\alpha}S_{\xi})$ is a compact set, and a space $X$ is pseudo-$\omega$-bounded if for each countable family $\mathcal{U}$ of non-empty open subsets of $X$, there exists a compact set $K\subseteq X$ such that each element in $\mathcal{U}$ has a 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 and only 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 and only if $X$ is $\alpha$-hyperbounded, for every infinite cardinal number $\alpha$.

Keywords: Hyperspaces; Vietoris topology; $\alpha$-hyperbounded spaces; pseudo-$\omega$-bounded spaces; normal and $C^*$-embedded spaces.

MSC: 54B20, 54D99, 54D15, 54C45

Pages:  306$-$318     

Volume  65 ,  Issue  3 ,  2013