$\mathfrak G$-bases in free (locally convex) topological vector spaces

TL;DR

This paper characterizes when free (locally convex) topological vector spaces have a local $rak G$-base, linking the base properties of the underlying space to the structure of the free space.

Contribution

It provides a new characterization of topological spaces whose free (locally convex) topological vector spaces possess a local $rak G$-base, using a novel description of their topology.

Findings

Identifies conditions for free topological vector spaces to have a local $rak G$-base.

Introduces a new approach to describe the topology of free topological vector spaces.

Connects base properties of the underlying space with the structure of the free space.

Abstract

We characterize topological (and uniform) spaces whose free (locally convex) topological vector spaces have a local $\mathfrak G$-base. A topological space $X$ has a local $\mathfrak G$-base if every point $x$ of $X$ has a neighborhood base $(U_\alpha)_{\alpha\in\omega^\omega}$ such that $U_\beta\subset U_\alpha$ for all $\alpha\le\beta$ in $\omega^\omega$. To construct $\mathfrak G$-bases in free topological vector spaces, we exploit a new description of the topology of a free topological vector space over a topological (or more generally, uniform) space.