Article
| Title: | On meager function spaces, network character and meager convergence in topological spaces (English) |
| Author: | Banakh, Taras |
| Author: | Mykhaylyuk, Volodymyr |
| Author: | Zdomskyy, Lyubomyr |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 52 |
| Issue: | 2 |
| Year: | 2011 |
| Pages: | 273-281 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | 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 $. |
| Keyword: | network character |
| Keyword: | meager convergent sequence |
| Keyword: | meager filter |
| Keyword: | meager space |
| Keyword: | function space |
| MSC: | 54A20 |
| MSC: | 54C35 |
| MSC: | 54E52 |
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| Date available: | 2011-05-17T08:40:20Z |
| Last updated: | 2012-08-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/141493 |
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