Lindelöf spaces C(X) over topological groups

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 σ-compact. (ii) Cp(X) is analytic. (iii) Cp(X) is K-analytic. (iv) Cp(X) is Lindelöf. (v) Cc(X) is a separable metrizable and complete locally convex space. (vi) Cc(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 Cp(X) is Lindelöf iff Cc(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.