# Free locally convex spaces with a small base Academic Article

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### 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.

### publication date

• June 27, 2016