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

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.

24 pages