On meager function spaces, network character and meager convergence in topological spaces

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 $\beta\omega$.