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Lindelöf spaces C(X) over topological groups

Kąkol, Jerzy and López Pellicer, Manuel and Martín Peinador, Elena and Tarieladze, Vaja (2008) Lindelöf spaces C(X) over topological groups. Forum Mathematicum, 20 (2). pp. 201-212. ISSN 0933-7741

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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 sigma-compact. (ii) C-p(X) is analytic. (iii) C-p(X) is K-analytic. (iv) C-p(X) is Lindelof. (v) C-c(X) is a separable metrizable and complete locally convex space. (vi) 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 Lindelof iff C-c(X) is weakly Lindelof. 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.

Item Type:Article
Uncontrolled Keywords:Locally convex-spaces; Banach-spaces; Compact-groups; Property; Sets
Subjects:Sciences > Mathematics > Topology
ID Code:12712
Deposited On:13 May 2011 11:14
Last Modified:06 Feb 2014 09:30

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