On topological spaces and topological groups with certain local countable networks

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.