Reflexivity in precompact groups and extensions

Abstract

We establish some general principles and find some counter-examples concerning the Pontryagin reflexivity of precompact groups and P-groups. We prove in particular that:; (1) A precompact Abelian group G of bounded order is reflexive if the dual group G<^> has no infinite compact subsets and every compact subset of G is contained in a compact subgroup of G.; (2) Any extension of a reflexive P-group by another reflexive P-group is again reflexive.; We show on the other hand that an extension of a compact group by a reflexive omega-bounded group (even dual to a reflexive P-group) can fail to be reflexive.; We also show that the P-modification of a reflexive sigma-compact group can be non-reflexive (even if, as proved in [20], the P-modification of a locally compact Abelian group is always reflexive). (C) 2013 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (published version

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

“…This proves the universality of ψ for S.A subgroup S of a topological abelian group G is said to be dually embedded in G if every continuous character of S extends to a continuous character of G. The next lemma is well known in the special case of Hausdorff topological groups[11, Lemma 2.2]. Every subgroup S of a precompact topological abelian group G is dually embedded in G.Proof.…”

The separable quotient problem for topological groups

“…Case 3. m = rank G is finite. By Theorem 8.22 (10) of [13], rank G 0 is finite, where G 0 denotes the connected component of the identity. By Corollary 8.24 (i) of [13], m = dim G = dim G 0 .…”

“…Let ϕ : G → T be a continuous homomorphism with |ϕ(G)| > 1. Then ϕ extends a continuous homomorphism ψ : T c → T (see [10,Lemma 2.2]). By Lemma 4.1, there exist pairwise distinct indices α 1 , .…”