Free locally convex spaces with a small base

Abstract: The paper studies the free locally convex space $L(X)$ over a Tychonoff space $X$. Since for infinite $X$ the space $L(X)$ is never metrizable (even not Fr\'echet-Urysohn), a possible applicable generalized metric property for $L(X)$ is welcome. We propose a concept (essentially weaker than first-countability) which is known under the name a $\mathfrak{G}$-base. A space $X$ has a {\em $\mathfrak{G}$-base} if for every $x\in X$ there is a base ${ U_\alpha : \alpha\in\mathbb{N}^\mathbb{N}}$ of neighborhoods at $x$ such that $U_\beta \subseteq U_\alpha$ whenever $\alpha\leq\beta$ for all $\alpha,\beta\in\mathbb{N}^\mathbb{N}$, where $\alpha=(\alpha(n)){n\in\mathbb{N}}\leq \beta=(\beta(n)){n\in\mathbb{N}}$ if $\alpha(n)\leq\beta(n)$ for all $n\in\mathbb{N}$. We show that if $X$ is an Ascoli $\sigma$-compact space, then $L(X)$ has a $\mathfrak{G}$-base if and only if $X$ admits an Ascoli uniformity $\mathcal{U}$ with a $\mathfrak{G}$-base. We prove that if $X$ is a $\sigma$-compact Ascoli space of $\mathbb{N}^\mathbb{N}$-uniformly compact type, then $L(X)$ has a $\mathfrak{G}$-base. As an application we show: (1) if $X$ is a metrizable space, then $L(X)$ has a $\mathfrak{G}$-base if and only if $X$ is $\sigma$-compact, and (2) if $X$ is a countable Ascoli space, then $L(X)$ has a $\mathfrak{G}$-base if and only if $X$ has a $\mathfrak{G}$-base.