A Note on Ultracomplete Hyperspaces

Abstract

A space X is called ultracomplete if $\beta X {\setminus } X$ is hemicompact. Ultracompleteness is stronger than Čech completeness and weaker than local compactness. For a given space Y, the hyperspace of non-empty compact subsets of Y endowed with the Vietoris topology is denoted by ${\mathcal {K}}(Y)$. It is well know that ${\mathcal {K}}(Y)$ is Čech complete (locally compact, compact) when X so is. The hyperspace ${\mathcal {K}}(Z)$ is not ultracomplete whenever Z is the ultracomplete space $[0,1]{\setminus } \{1/n :n\in {\mathbb {N}} \}$. A space is $\omega$-hyperbounded if the closure of any $\sigma$-compact subspace is compact. In this work it is proved that ${\mathcal {K}}(X^\omega )$ is ultracomplete, if X is an $\omega$-hyperbounded locally compact space. It is also proved that ${\mathcal {K}}((X\setminus A )^\omega )$ is ultracomplete countably compact, whenever X is a compact space and A is a countable set containing only P-points of X.