Abstract:For a non-isolated point $x$ of a topological space $X$ let $\mathrm{nw}_\chi (x)$ be the smallest cardinality of a family $\mathcal N$ of infinite subsets of $X$ such that each neighborhood $O(x)\subset X$ of $x$ contains a set $N\in \mathcal N$. We prove that \begin{itemize} \leftskip= 25pt \rightskip= \leftskip \item each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with $\mathrm{nw}_\chi (x)=\aleph_0$; \item for each point $x\in X$ with $\mathrm{nw}_\chi (x)=\aleph_0$ there is an injective sequence $(x_n)_{n\in \omega }$ in $X$ that $\mathcal F$-converges to $x$ for some meager filter $\mathcal F$ on $\omega $; \item if a functionally Hausdorff space $X$ contains an $\mathcal F$-convergent injective sequence for some meager filter $\mathcal F$, then for every path-connected space $Y$ that contains two non-empty open sets with disjoint closures, the function space $C_p(X,Y)$ is meager. \end{itemize} Also we investigate properties of filters $\mathcal F$ admitting an injective $\mathcal F$-convergent sequence in $\beta \omega $.
Keywords: network character, meager convergent sequence, meager filter, meager space, function space
AMS Subject Classification: 54A20 54C35 54E52