  For a non-isolated point $x$ of a topological space $X$ the network character
$nw_\chi(x)$ is the smallest cardinality of a family of infinite subsets of $X$
such that each neighborhood $O(x)$ of $x$ contains a set from the family. We
prove that (1) each infinite compact Hausdorff space $X$ contains a
non-isolated point $x$ with $nw_\chi(x)=\aleph_0$; (2) for each point $x\in X$
with countable character there is an injective sequence in $X$ that
$\F$-converges to $x$ for some meager filter $\F$ on $\omega$; (3) if a
functionally Hausdorff space $X$ contains an $\F$-convergent injective sequence
for some meager filter $\F$, then for every $T_1$-space $Y$ that contains two
non-empty open sets with disjoint closures, the function space $C_p(X,Y)$ is
meager. Also we investigate properties of filters $\F$ admitting an injective
$\F$-convergent sequence in $\beta\omega$.
