When is an ultracomplete space almost locally compact?

Daniel Jardón Arcos, Vladimir V. Tkachuk

Abstract

We study spaces X which have a countable outer base in βX; they are called ultracomplete in the most recent terminology. Ultracompleteness implies Cech-completeness and is implied by almost local compactness (≡having all points of non-local compactness inside a compact subset of countable outer character). It turns out that ultracompleteness coincides with almost local compactness in most important classes of isocompact spaces (i.e., in spaces in which every countably compact subspace is compact). We prove that if an isocompact space X is ω-monolithic then any ultracomplete subspace of X is almost locally compact. In particular, any ultracomplete subspace of a compact ω-monolithic space of countable tightness is almost locally compact. Another consequence of this result is that, for any space X such that vX is a Lindelöf Σ-space, a subspace of Cp(X) is ultracomplete if and only if it is almost locally compact. We show that it is consistent with ZFC that not all ultracomplete subspaces of hereditarily separable compact spaces are almost locally compact.


Keywords

Ultracompleteness; Cech-completeness; Countable type; Pointwise countable type; Lindelöf Σ-spaces; Splittable spaces; Eberlein compact spaces; Almost locally compact spaces; Isocompact spaces

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References

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1. Ultracompleteness of hyperspaces of compact sets
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Acta Mathematica Hungarica  vol: 137  issue: 1-2  first page: 139  year: 2012  
doi: 10.1007/s10474-012-0223-6



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