Abstract
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 F n (X)={K∈2X∣K has at most n points} is also countably compact ultracomplete for every natural number n. We also analyse ultracompleteness of F n (X) and 2X.
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Research supported by Consejo Nacional de Ciencia y Tecnología (CONACyT) and UACM.
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Jardón, D. Ultracompleteness of hyperspaces of compact sets. Acta Math Hung 137, 139–152 (2012). https://doi.org/10.1007/s10474-012-0223-6
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DOI: https://doi.org/10.1007/s10474-012-0223-6
Key words and phrases
- ultracomplete space
- Čech-complete space
- compactification
- countable type
- almost locally compact space
- countably compact space
- hemicompact space
- hyperspace of compact sets
- hyperspace of finite sets
- Vietoris topology
- symmetric product