On topological groups admitting a base at the identity indexed by $\omega ^{\omega }$

Arkady G. Leiderman¹ ; Vladimir G. Pestov² ; Artur H. Tomita³

¹ Department of Mathematics Ben-Gurion University of the Negev P.O.B. 653 Beer Sheva, Israel
² Department of Mathematics and Statistics University of Ottawa 585 King Edward Avenue Ottawa, Ontario K1N 6N5, Canada and Departamento de Matemática Universidade Federal de Santa Catarina Trindade, Florianópolis, SC, 88.040-900, Brazil
³ Instituto de Matemática e Estatistica Universidade de São Paulo Rua do Matao, 1010 CEP 05508-090, São Paulo, Brazil

Fundamenta Mathematicae, Tome 238 (2017) no. 1, pp. 79-100.

Résumé

A topological group $G$ is said to have a local $\omega ^\omega $-base if the neighbourhood system at the identity admits a monotone cofinal map from the directed set $\omega ^\omega $. In particular, every metrizable group is such, but the class of groups with a local $\omega ^\omega $-base is significantly wider. The aim of this article is to better understand the boundaries of this class, by presenting new examples and counter-examples. Ultraproducts and non-archimedean ordered fields lead to natural families of non-metrizable groups with a local $\omega ^\omega $-base which nevertheless are Baire topological spaces. More examples come from such constructions as the free topological group $F(X)$ and the free Abelian topological group $A(X)$ of a Tychonoff (more generally uniform) space $X$, as well as the free product of topological groups. We show that 1) the free product of countably many separable topological groups with a local $\omega ^\omega $-base admits a local $\omega ^\omega $-base; 2) the group $A(X)$ of a Tychonoff space $X$ admits a local $\omega ^\omega $-base if and only if the finest uniformity of $X$ has an $\omega ^\omega $-base; 3) the group $F(X)$ of a Tychonoff space $X$ admits a local $\omega ^\omega $-base provided $X$ is separable and the finest uniformity of $X$ has an $\omega ^\omega $-base.

Détail

DOI : 10.4064/fm188-9-2016

Keywords: topological group said have local omega omega base neighbourhood system identity admits monotone cofinal map directed set omega omega particular every metrizable group class groups local omega omega base significantly wider article better understand boundaries class presenting examples counter examples ultraproducts non archimedean ordered fields lead natural families non metrizable groups local omega omega base which nevertheless baire topological spaces examples come constructions topological group abelian topological group tychonoff generally uniform space nbsp product topological groups product countably many separable topological groups local omega omega base admits local omega omega base group tychonoff space admits local omega omega base only finest uniformity has omega omega base group tychonoff space admits local omega omega base provided separable finest uniformity has omega omega base