| 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 |
| . |
| Category:
|
math |
| . |
| 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 $. (English) |
| Keyword:
|
network character |
| Keyword:
|
meager convergent sequence |
| Keyword:
|
meager filter |
| Keyword:
|
meager space |
| Keyword:
|
function space |
| MSC:
|
54A20 |
| MSC:
|
54C35 |
| MSC:
|
54E52 |
| idZBL:
|
Zbl 1240.54018 |
| idMR:
|
MR2849049 |
| . |
| Date available:
|
2011-05-17T08:40:20Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141493 |
| . |
| Reference:
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