Geodesic
On meager function spaces, network character and meager convergence in topological spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 2, pp. 273-281 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $.
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 $.
Classification : 54A20, 54C35, 54E52
Keywords: network character; meager convergent sequence; meager filter; meager space; function space 
@article{CMUC_2011_52_2_a6,
     author = {Banakh, Taras and Mykhaylyuk, Volodymyr and Zdomskyy, Lyubomyr},
     title = {On meager function spaces, network character and meager convergence in topological spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {273--281},
     year = {2011},
     volume = {52},
     number = {2},
     mrnumber = {2849049},
     zbl = {1240.54018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2011_52_2_a6/}
}
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Banakh, Taras; Mykhaylyuk, Volodymyr; Zdomskyy, Lyubomyr. On meager function spaces, network character and meager convergence in topological spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 2, pp. 273-281. http://geodesic.mathdoc.fr/item/CMUC_2011_52_2_a6/