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

TL;DR

This paper characterizes topological and uniform spaces whose free topological algebraic structures, like vector spaces and groups, possess an $oldsymbol{ ext{ extomega}}^ ext{ extomega}$-base, linking space properties to algebraic free constructions.

Contribution

It provides a characterization of spaces with $oldsymbol{ ext{ extomega}}^ ext{ extomega}$-bases whose free algebraic structures also have such bases, advancing understanding of base properties in free topological objects.

Findings

Identifies conditions for free topological vector spaces to have $ ext{ extomega}^ ext{ extomega}$-bases.

Characterizes when free topological groups possess $ ext{ extomega}^ extomega$-bases.

Links base properties of spaces to those of their free algebraic structures.

Abstract

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.