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

Taras O. Banakh; Volodymyr Mykhaylyuk; Lubomyr Zdomsky

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 2, page 273-281
  • ISSN: 0010-2628

Abstract

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For a non-isolated point x of a topological space X let nw χ ( x ) be the smallest cardinality of a family 𝒩 of infinite subsets of X such that each neighborhood O ( x ) ⊂ X of x contains a set N ∈ 𝒩 . We prove that (a) each infinite compact Hausdorff space X contains a non-isolated point x with nw χ ( x ) = ℵ 0 ; (b) for each point x ∈ X with nw χ ( x ) = ℵ 0 there is an injective sequence ( x n ) n ∈ ω in X that ℱ -converges to x for some meager filter ℱ on ω ; (c) if a functionally Hausdorff space X contains an ℱ -convergent injective sequence for some meager filter 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 ℱ admitting an injective ℱ -convergent sequence in β ω .

How to cite

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Banakh, Taras O., Mykhaylyuk, Volodymyr, and Zdomsky, Lubomyr. "On meager function spaces, network character and meager convergence in topological spaces." Commentationes Mathematicae Universitatis Carolinae 52.2 (2011): 273-281. <http://eudml.org/doc/246994>.

@article{Banakh2011,
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 (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 $.},
author = {Banakh, Taras O., Mykhaylyuk, Volodymyr, Zdomsky, Lubomyr},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {network character; meager convergent sequence; meager filter; meager space; function space; network character; meager convergent sequence; meager filter; meager space; function space},
language = {eng},
number = {2},
pages = {273-281},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On meager function spaces, network character and meager convergence in topological spaces},
url = {http://eudml.org/doc/246994},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Banakh, Taras O.
AU - Mykhaylyuk, Volodymyr
AU - Zdomsky, Lubomyr
TI - On meager function spaces, network character and meager convergence in topological spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 2
SP - 273
EP - 281
AB - 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 $.
LA - eng
KW - network character; meager convergent sequence; meager filter; meager space; function space; network character; meager convergent sequence; meager filter; meager space; function space
UR - http://eudml.org/doc/246994
ER -

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