Free locally convex spaces with a small base

0474089 - MÚ 2018 RIV ES eng J - Journal Article
Gabriyelyan, S. - Kąkol, Jerzy
Free locally convex spaces with a small base.
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Roč. 111, č. 2 (2017), s. 575-585. ISSN 1578-7303. E-ISSN 1579-1505
R&D Projects: GA ČR GF16-34860L
Institutional support: RVO:67985840
Keywords : compact resolution * free locally convex space * G-base
OECD category: Pure mathematics
Impact factor: 1.074, year: 2017
http://link.springer.com/article/10.1007%2Fs13398-016-0315-1

The paper studies the free locally convex space L(X) over a Tychonoff space X. Since for infinite X the space L(X) is never metrizable (even not Fréchet-Urysohn), a possible applicable generalized metric property for L(X) is welcome. We propose a concept (essentially weaker than first-countability) which is known under the name a G-base. A space X has a G-base if for every x \in X there is a base { U\alpha: \alpha \in NN} of neighborhoods at x such that Uta ... U\alpha whenever \alpha ... ta for all \alpha, ta \in NN, where \alpha = (\alpha(n)) n \in N... N. We show that if X is an Ascoli \omega-compact space, then L(X) has a G-base if and only if X admits an Ascoli uniformity U with a G-base. We prove that if X is a \omega-compact Ascoli space of NN-uniformly compact type, then L(X) has a G-base. As an application we show: (1) if X is a metrizable space, then L(X) has a G-base if and only if X is \omega-compact, and (2) if X is a countable Ascoli space, then L(X) has a G-base if and only if X has a G-base.
Permanent Link: http://hdl.handle.net/11104/0271206