When is an ultracomplete space almost locally compact?
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 char...
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2006-10-01
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doaj-8d39ca5a1f8a47f5b27626088d8c4c012020-11-24T23:13:32ZengUniversitat Politècnica de ValènciaApplied General Topology1576-94021989-41472006-10-017219120110.4995/agt.2006.19231548When is an ultracomplete space almost locally compact?Daniel Jardón Arcos0Vladimir V. Tkachuk1Universidad Autónoma MetropolitanaUniversidad Autónoma MetropolitanaWe 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.http://polipapers.upv.es/index.php/AGT/article/view/1923UltracompletenessCech-completenessCountable typePointwise countable typeLindelöf Σ-spacesSplittable spacesEberlein compact spacesAlmost locally compact spacesIsocompact spaces |
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DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Daniel Jardón Arcos Vladimir V. Tkachuk |
spellingShingle |
Daniel Jardón Arcos Vladimir V. Tkachuk When is an ultracomplete space almost locally compact? Applied General Topology Ultracompleteness Cech-completeness Countable type Pointwise countable type Lindelöf Σ-spaces Splittable spaces Eberlein compact spaces Almost locally compact spaces Isocompact spaces |
author_facet |
Daniel Jardón Arcos Vladimir V. Tkachuk |
author_sort |
Daniel Jardón Arcos |
title |
When is an ultracomplete space almost locally compact? |
title_short |
When is an ultracomplete space almost locally compact? |
title_full |
When is an ultracomplete space almost locally compact? |
title_fullStr |
When is an ultracomplete space almost locally compact? |
title_full_unstemmed |
When is an ultracomplete space almost locally compact? |
title_sort |
when is an ultracomplete space almost locally compact? |
publisher |
Universitat Politècnica de València |
series |
Applied General Topology |
issn |
1576-9402 1989-4147 |
publishDate |
2006-10-01 |
description |
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. |
topic |
Ultracompleteness Cech-completeness Countable type Pointwise countable type Lindelöf Σ-spaces Splittable spaces Eberlein compact spaces Almost locally compact spaces Isocompact spaces |
url |
http://polipapers.upv.es/index.php/AGT/article/view/1923 |
work_keys_str_mv |
AT danieljardonarcos whenisanultracompletespacealmostlocallycompact AT vladimirvtkachuk whenisanultracompletespacealmostlocallycompact |
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