On topological groups admitting a base at identity indexed with $\omega^\omega$
| Autor: | Leiderman, Arkady G., Pestov, Vladimir G., Tomita, Artur H. |
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| Rok vydání: | 2015 |
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| Zdroj: | Fund. Math. 238 (2017), no. 1, 79-100 |
| Druh dokumentu: | Working Paper |
| Popis: | A topological group $G$ is said to have a local $\omega^\omega$-base if the neighbourhood system at 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-arichimedean 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 a $\omega^\omega$-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 a $\omega^\omega$-base. Comment: 20 pages, latex 2e, accepted for publication in Fundam. Math |
| Databáze: | arXiv |
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