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 Fréchet-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 $\mathfrak {G}$-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 $\alpha \le \beta$ for all $\alpha ,\beta \in \mathbb {N}^\mathbb {N}$, where $\alpha =(\alpha (n))_{n\in \mathbb {N}}\le \beta =(\beta (n))_{n\in \mathbb {N}}$ if $\alpha (n)\le \beta (n)$ 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.