## Abstract

A topological group G is said to have a local ^{w}-base if the neighbourhood system at the identity admits a monotone cofinal map from the directed set ^{w}. In particular, every metrizable group is such, but the class of groups with a local ^{w}-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 nonarchimedean ordered fields lead to natural families of non-metrizable groups with a local ^{w}-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 ^{w}-base admits a local ^{w}-base; 2) the group A(X) of a Tychonoff space X admits a local ^{w}-base if and only if the finest uniformity of X has an ^{w}-base; 3) the group F(X) of a Tychonoff space X admits a local ^{w}-base provided X is separable and the finest uniformity of X has an ^{w}-base.

Original language | English |
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Pages (from-to) | 79-100 |

Number of pages | 22 |

Journal | Fundamenta Mathematicae |

Volume | 238 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2017 |

## Keywords

- Free topological group
- Topological group
- Tukey order
- Uniform space

## ASJC Scopus subject areas

- Algebra and Number Theory

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