A class of angelic sequential Fréchet-Urysohn topological groups

Abstract

Fréchet–Urysohn (briefly F-U) property for topological spaces is known to be highly non-multiplicative; for instance, the square of a compact F-U space is not in general Fréchet–Urysohn [P. Simon, A compact Fréchet space whose square is not Fréchet, Comment.Math. Univ. Carolin. 21 (1980) 749–753. [27]]. Van Douwen proved that the product of a metrizable space by a Fréchet– Urysohn space may not be (even) sequential. If the second factor is a topological group this behaviour improves significantly: we have obtained (Theorem 1.6(c)) that the product of a first countable space by a F-U topological group is a F-U space. We draw some important consequences by interacting this fact with Pontryagin duality theory. The main results are the following: (1) If the dual group of a metrizable Abelian group is F-U, then it must be metrizable and locally compact. (2) Leaning on (1) we point out a big class of hemicompact sequential non-Fréchet–Urysohn groups, namely: the dual groups of metrizable separable locally quasi-convex non-locally precompact groups. The members of this class are furthermore complete, strictly angelic and locally quasi-convex. (3) Similar results are also obtained in the framework of locally convex spaces.