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

Banakh, Taras ; Mykhaylyuk, Volodymyr ; Zdomskyy, Lyubomyr

Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 2, pp. 273-281.

Résumé

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 (a) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with $\mathrm{nw}_\chi (x)=\aleph_0$; (b) 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 $; (c) 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. Also we investigate properties of filters $\mathcal F$ admitting an injective $\mathcal F$-convergent sequence in $\beta \omega $.

Détail

Classification : 54A20, 54C35, 54E52
Keywords: network character; meager convergent sequence; meager filter; meager space; function space