Sequential properties of function spaces with the compact-open topology

Gary Gruenhage; Boaz Tsaban; Lyubomyr Zdomskyy

doi:10.1016/j.topol.2010.11.012

Sequential properties of function spaces with the compact-open topology

Gary Gruenhage, Boaz Tsaban, Lyubomyr Zdomskyy

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

The main results of the paper are:. (1)If X is metrizable but not locally compact topological space, then Ck(X) contains a closed copy of S₂, and hence does not have the property AP;(2)For any zero-dimensional Polish X, the space C_k(X,2) is sequential if and only if X is either locally compact or the derived set X×, is compact; and(3)All spaces of the form C_k(X,2), where X is a non-locally compact Polish space whose derived set is compact, are homeomorphic, and have the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the (n+1)st.

Original language	English
Pages (from-to)	387-391
Number of pages	5
Journal	Topology and its Applications
Volume	158
Issue number	3
DOIs	https://doi.org/10.1016/j.topol.2010.11.012
State	Published - 15 Feb 2011

Bibliographical note

Funding Information:
✩ The third named author acknowledges the support of the FWF grant P19898-N18. We thank the referees for comments and suggestions.

Keywords

AP
Arens space
Compact-open topology
Fréchet-Urysohn
Polish space
Pytkeev property
Sequential
Sequential fan
WAP

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10.1016/j.topol.2010.11.012

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abstract = "The main results of the paper are:. (1)If X is metrizable but not locally compact topological space, then Ck(X) contains a closed copy of S2, and hence does not have the property AP;(2)For any zero-dimensional Polish X, the space Ck(X,2) is sequential if and only if X is either locally compact or the derived set X×, is compact; and(3)All spaces of the form Ck(X,2), where X is a non-locally compact Polish space whose derived set is compact, are homeomorphic, and have the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the (n+1)st.",

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author = "Gary Gruenhage and Boaz Tsaban and Lyubomyr Zdomskyy",

note = "Funding Information: ✩ The third named author acknowledges the support of the FWF grant P19898-N18. We thank the referees for comments and suggestions.",

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AU - Zdomskyy, Lyubomyr

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KW - Arens space

KW - Compact-open topology

KW - Fréchet-Urysohn

KW - Polish space

KW - Pytkeev property

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KW - Sequential fan

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JO - Topology and its Applications

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ER -

Sequential properties of function spaces with the compact-open topology

Abstract

Bibliographical note

Keywords

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