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 Frechet-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.