[logic-ml] Kobe Colloquium on Logic, Statistics and Informatics, August 21 (Arkady Leiderman)

[Joerg Brendle]

Kobe Colloquium on Logic, Statistics and Informatics

以下の要領でコロキウムを開催します。

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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交通：阪急六甲駅またはJR六甲道駅から神戸市バス36系統「鶴甲団地」
行きに乗車，「神大本部工学部前」停留所下車，徒歩すぐ．
http://www.kobe-u.ac.jp/info/access/rokko/rokkodai-dai2.htm
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