Is a monotone union of contractible open sets contractible?

This paper presents some partial answers to the following question. QuestionIf a normal space X is the union of an increasing sequence of open sets U1⊂U2⊂U3⊂… such that each Un contracts to a point in X, must X be contractible? The main results of the paper are: Theorem 1If a normal space X is the u...

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Bibliographic Details
Published inTopology and its applications Vol. 214; pp. 89 - 93
Main Authors Ancel, Fredric D., Edwards, Robert D.
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.12.2016
Subjects
Contractibiity
Locally compact
Monotone union
Normal space
σ-Compact
55P99
Normal space
54D99
55M99
57N99
Monotone union
σ-Compact
Contractibiity
Locally compact
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Summary:This paper presents some partial answers to the following question. QuestionIf a normal space X is the union of an increasing sequence of open sets U1⊂U2⊂U3⊂… such that each Un contracts to a point in X, must X be contractible? The main results of the paper are: Theorem 1If a normal space X is the union of a sequence of open subsets{Un}such thatcl(Un)⊂Un+1andUncontracts to a point inUn+1for eachn≥1, then X is contractible. Corollary 2If a locally compact σ-compact normal space X is the union of an increasing sequence of open setsU1⊂U2⊂U3⊂…such that eachUncontracts to a point in X, then X is contractible.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2016.10.001