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.

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Correspondence to Saak Gabriyelyan.

Additional information

The research was supported for Jerzy Ka̧kol by Generalitat Valenciana, Conselleria d’Educació i Esport, Spain, Grant PROMETEO/2015/058 and by the GAČR project 16-34860L and RVO: 67985840. Jerzy Ka̧kol gratefully acknowledges also the financial support he received from the Center for Advanced Studies in Mathematics of the Ben Gurion University of the Negev during his visit March 15–22, 2016.

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Gabriyelyan, S., Ka̧kol, J. Free locally convex spaces with a small base. RACSAM 111, 575–585 (2017). https://doi.org/10.1007/s13398-016-0315-1

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Keywords

  • Free locally convex space
  • \(\mathfrak {G}\)-base
  • \(C_k(X)\)
  • Compact resolution