A class of angelic sequential non-Fréchet-Urysohn topological groups

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.

Another class of sequential non-Fréchet-Urysohn complete topological Abelian groups very different from ours is given in E.G. Zelenyuk, I.V. Protasov, Topologies of Abelian groups, Math. USSR Izv. 37 (2) (1991) 445-460. 32]].