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Lindelöf spaces C ( X ) over topological groups

Lindelöf spaces C ( X ) over topological groups Theorem 1 proves (among the others) that for a locally compact topological group X the following assertions are equivalent: (i) X is metrizable and ॣ-compact. (ii) C p ( X ) is analytic. (iii) C p ( X ) is K -analytic. (iv) C p ( X ) is Lindelöf. (v) C c ( X ) is a separable metrizable and complete locally convex space. (vi) C c ( X ) is compactly dominated by irrationals . This result supplements earlier results of Corson, Christensen and Calbrix and provides several applications, for example, it easily applies to show that: (1) For a compact topological group X the Eberlein, Talagrand, Gul'ko and Corson compactness are equivalent and any compact group of this type is metrizable. (2) For a locally compact topological group X the space C p ( X ) is Lindelöf iff C c ( X ) is weakly Lindelöf. The proofs heavily depend on the following result of independent interest: A locally compact topological group X is metrizable iff every compact subgroup of X has countable tightness (Theorem 2). More applications of Theorem 1 and Theorem 2 are provided. 1991 Mathematics Subject Classification: 22A05, 43A40, 54H11. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Lindelöf spaces C ( X ) over topological groups

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Publisher
de Gruyter
Copyright
© Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/FORUM.2008.010
Publisher site
See Article on Publisher Site

Abstract

Theorem 1 proves (among the others) that for a locally compact topological group X the following assertions are equivalent: (i) X is metrizable and ॣ-compact. (ii) C p ( X ) is analytic. (iii) C p ( X ) is K -analytic. (iv) C p ( X ) is Lindelöf. (v) C c ( X ) is a separable metrizable and complete locally convex space. (vi) C c ( X ) is compactly dominated by irrationals . This result supplements earlier results of Corson, Christensen and Calbrix and provides several applications, for example, it easily applies to show that: (1) For a compact topological group X the Eberlein, Talagrand, Gul'ko and Corson compactness are equivalent and any compact group of this type is metrizable. (2) For a locally compact topological group X the space C p ( X ) is Lindelöf iff C c ( X ) is weakly Lindelöf. The proofs heavily depend on the following result of independent interest: A locally compact topological group X is metrizable iff every compact subgroup of X has countable tightness (Theorem 2). More applications of Theorem 1 and Theorem 2 are provided. 1991 Mathematics Subject Classification: 22A05, 43A40, 54H11.

Journal

Forum Mathematicum – de Gruyter

Published: Mar 1, 2008

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