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oai:docta.ucm.es:20.500.14352/501342023-08-26T17:04:12Zcom_20.500.14352_14col_20.500.14352_15

A class of angelic sequential non-Frechet-Urysohn topological groups Chasco, M.J. Martín Peinador, Elena Tarieladze, Vaja Feechet-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 Frechet-Urysohn [P. Simon, A compact Frechet space whose square is not Frechet, Comment. Math. Univ. Carolin. 21 (1980) 749-753. [27]]. Van Douwen proved that the product of a metrizable space by a Frechet-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-Frechet-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-Frechet-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]]. 2023-06-20T09:39:41Z 2023-06-20T09:39:41Z 2023-06-20T09:39:41Z 2007-02-01 journal article https://hdl.handle.net/20.500.14352/50134 0166-8641 10.1016/j.topol.2006.08.008 eng restricted access Elsevier Science