On topological spaces and topological groups with certain local countable networks

TL;DR

This paper investigates the local topological properties of spaces and groups with countable networks, characterizing metrizability and other properties, and applies these results to spaces of distributions, free topological groups, and locally convex spaces.

Contribution

It introduces new characterizations of spaces with countable $cp$- and $cn$-networks, linking them to metrizability, separability, and other structural properties, with applications to various functional spaces.

Findings

A Baire topological group is metrizable iff it has the strong Pytkeev property.

A topological group has a countable $cp$-network iff it is separable with such a network at the unit.

The space of distributions $D'(\Omega)$ has a countable $cp$-network, improving known tightness results.

Abstract

Being motivated by the study of the space $C_c(X)$ of all continuous real-valued functions on a Tychonoff space $X$ with the compact-open topology, we introduced in [15] the concepts of a $cp$-network and a $cn$-network (at a point $x$) in $X$. In the present paper we describe the topology of $X$ admitting a countable $cp$- or $cn$-network at a point $x\in X$. This description applies to provide new results about the strong Pytkeev property, already well recognized and applicable concept originally introduced by Tsaban and Zdomskyy [43]. We show that a Baire topological group $G$ is metrizable if and only if $G$ has the strong Pytkeev property. We prove also that a topological group $G$ has a countable $cp$-network if and only if $G$ is separable and has a countable $cp$-network at the unit. As an application we show, among the others, that the space $D'(\Omega)$ of distributions over open $\Omega\subseteq\mathbb{R}^{n}$ has a countable $cp$-network, which essentially improves the well known fact stating that $D'(\Omega)$ has countable tightness. We show that, if $X$ is an $\mathcal{MK}_\omega$-space, then the free topological group $F(X)$ and the free locally convex space $L(X)$ have a countable \mbox{$cp$-network}. We prove that a topological vector space $E$ is $p$-normed (for some \mbox{$0<p\leq 1$}) if and only if $E$ is Fr\'echet-Urysohn and admits a fundamental sequence of bounded sets.