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 s-compact. (ii) CpðXÞ is analytic. (iii) CpðXÞ is K-analytic. (iv) CpðXÞ is Lindelöf. (v) CcðX Þ is a separable metrizable and complete locally convex space. (vi) CcðX Þ 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 CpðX Þ is Lindelöf i¤ CcðXÞ is weakly Lindelöf. The proofs heavily depend on the following result of independent interest: A locally compact topological group X is metrizable i¤ 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.