  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.
