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  • 1 Universidad Autónoma de la Ciudad de México (UACM), Mexico City, Mexico
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A space X is called ultracomplete if it has countable character in its Stone–Čech compactification βX. A space X is called almost locally compact if the set of all points at which X is not locally compact is contained in a compact set of countable outer character. For a given Tychonoff space X let 2X be the hyperspace of all nonempty compact subsets of X endowed with the Vietoris topology. We prove that 2X is almost locally compact if and only if X is locally compact. We also prove that for a countably compact ultracomplete space X the hyperspace Fn(X)={K∊2XK has at most n points} is also countably compact ultracomplete for every natural number n. We also analyse ultracompleteness of Fn(X) and 2X.

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