Kobe Colloquium on Logic, Statistics and Informatics

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日時：2015年8月21日（金）16:00-17:00 講演者：Arkady Leiderman （ベングリオン大学） 場所：神戸大学自然科学総合研究棟3号館4階421室（プレゼンテーション室）

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題目： Open G-bases and compact resolutions in topological groups and locally convex spaces.

アブストラクト： A (Hausdorff) topological group G is said to have a {G}-base if G admits a base of neighbourhoods of the unit {U_alpha: alpha in N^N} such that U_alpha is contained in U_beta whenever beta leq alpha for all alpha, beta in N^N.

The class of all metrizable topological groups is a proper subclass of the class TG_{G} of all topological groups having a {G}-base. A relation to the known combinatorial cardinal invariants b and d has been established: If a topological group G is in TG_{G}, then chi(G) in { 1, aleph_0 } cup [b,d]. We prove that a topological group G is metrizable iff G is Fréchet-Urysohn and has a {G}-base.

We also show that any precompact set in a topological group G in TG_{G} is metrizable, and hence G is strictly angelic. We deduce from this result that an almost metrizable group G is metrizable iff G has a {G}-base.

Characterizations of metrizability of topological vector spaces, in particular C_c(X), are given using {G}-bases. We obtain a result stating that if X is a submetrizable k_omega-space, then the free abelian topological group A(X) and the free locally convex topological space L(X) have a {G}-base. Another class TG_CR of topological groups with a compact resolution swallowing the compact sets appears naturally in this article. We show that the classes TG_CR and TG_{G} in some sense are dual to each other.

We show also that the strong Pytkeev property for general topological groups is closely related to the notion of a {G}-base. We pose a dozen open questions.

References:

1) On topological groups with a small base and metrizability, Saak Gabriyelyan, Jerzy Kąkol and Arkady Leiderman, Fund. Math. 229 (2015), 129-158.

2) The strong Pytkeev property for topological groups and topological vector spaces*,* S. S. Gabriyelyan , J. Ka̧kol and A. Leiderman, Monatshefte für Mathematik, December 2014, Volume 175, Issue 4, pp 519-542. ========================================================

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