$ω^ω$-Dominated function spaces and $ω^ω$-bases in free objects of Topological Algebra

Taras Banakh, Arkady Leiderman

Published 2016-11-19Version 1

A topological space $X$ is defined to have an $\omega^\omega$-base} if at each point $x\in X$ the space $X$ has a neighborhood base $(U_\alpha[x])_{\alpha\in\omega^\omega}$ such that $U_\beta[x]\subset U_\alpha[x]$ for all $\alpha\le\beta$ in $\omega^\omega$. We characterize topological and uniform spaces whose free (locally convex) topological vector spaces or free (Abelian or Boolean) topological groups have $\omega^\omega$-bases.