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- W2338603167 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 $xin X$ there is a base ${ U_alpha : alphainmathbb{N}^mathbb{N}}$ of neighborhoods at $x$ such that $U_beta subseteq U_alpha$ whenever $alphaleqbeta$ for all $alpha,betainmathbb{N}^mathbb{N}$, where $alpha=(alpha(n))_{ninmathbb{N}}leq beta=(beta(n))_{ninmathbb{N}}$ if $alpha(n)leqbeta(n)$ for all $ninmathbb{N}$. 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." @default.
- W2338603167 created "2016-06-24" @default.
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- W2338603167 date "2016-04-09" @default.
- W2338603167 modified "2023-10-18" @default.
- W2338603167 title "Free locally convex spaces with a small base" @default.
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