A Note on Ultracomplete Hyperspaces

A space X is called ultracomplete if βX\X is hemicompact. Ultracompleteness is stronger than Cech completeness and weaker than local compactness. For a given space Y , the hyperspace of non-empty compact subsets of Y endowedwith theVietoris topology is denoted by K(Y ). It is well know that K(Y ) is Cech complete (locally compact, compact)when X so is. The hyperspace K(Z) is not ultra complete when ever Z is the ultracomplete space [0, 1]\{1/n : n ∈ N}. A space is ω-hyperbounded if the closure of any σ-compact subspace is compact. In this work it is proved that K(Xω) is ultracomplete, if X is an ω-hyperbounded locally compact space. It is also proved that K((X \ A)ω) is ultracomplete countably compact, whenever X is a compact space and A is a countable set containing only P-points of X.