𝒫 -approximable compact spaces

Mihail G. Tkachenko

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 3, page 583-595
  • ISSN: 0010-2628

Abstract

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For every topological property 𝒫 , we define the class of 𝒫 -approximable spaces which consists of spaces X having a countable closed cover γ such that the “section” X ( x , γ ) = ⋂ { F ∈ γ : x ∈ F } has the property 𝒫 for each x ∈ X . It is shown that every 𝒫 -approximable compact space has 𝒫 , if 𝒫 is one of the following properties: countable tightness, ℵ 0 -scatteredness with respect to character, C -closedness, sequentiality (the last holds under MA or 2 ℵ 0 < 2 ℵ 1 ). Metrizable-approximable spaces are studied: every compact space in this class has a dense, Čech-complete, paracompact subspace; moreover, if X is linearly ordered, then X contains a dense metrizable subspace.

How to cite

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Tkachenko, Mihail G.. "$\mathcal {P}$-approximable compact spaces." Commentationes Mathematicae Universitatis Carolinae 32.3 (1991): 583-595. <http://eudml.org/doc/247317>.

@article{Tkachenko1991,
abstract = {For every topological property $\mathcal \{P\}$, we define the class of $\mathcal \{P\}$-approximable spaces which consists of spaces X having a countable closed cover $\gamma $ such that the “section” $X(x,\gamma )= \bigcap \lbrace F\in \gamma :x\in F\rbrace $ has the property $\mathcal \{P\}$ for each $x\in X$. It is shown that every $\mathcal \{P\}$-approximable compact space has $\mathcal \{P\}$, if $\mathcal \{P\}$ is one of the following properties: countable tightness, $\aleph _0$-scatteredness with respect to character, $C$-closedness, sequentiality (the last holds under MA or $2^\{\aleph _0\}<2^\{\aleph _1\}$). Metrizable-approximable spaces are studied: every compact space in this class has a dense, Čech-complete, paracompact subspace; moreover, if $X$ is linearly ordered, then $X$ contains a dense metrizable subspace.},
author = {Tkachenko, Mihail G.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$\mathcal \{P\}$-approximable space; Lindelöf $\Sigma $-space; compact; metrizable; $C$-closed; sequential; linearly ordered; Lindelöf -space; metrizable-approximable spaces; - approximable spaces; countable tightness; -scatteredness; compact space; dense, Čech-complete paracompact subspace},
language = {eng},
number = {3},
pages = {583-595},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {$\mathcal \{P\}$-approximable compact spaces},
url = {http://eudml.org/doc/247317},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Tkachenko, Mihail G.
TI - $\mathcal {P}$-approximable compact spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 3
SP - 583
EP - 595
AB - For every topological property $\mathcal {P}$, we define the class of $\mathcal {P}$-approximable spaces which consists of spaces X having a countable closed cover $\gamma $ such that the “section” $X(x,\gamma )= \bigcap \lbrace F\in \gamma :x\in F\rbrace $ has the property $\mathcal {P}$ for each $x\in X$. It is shown that every $\mathcal {P}$-approximable compact space has $\mathcal {P}$, if $\mathcal {P}$ is one of the following properties: countable tightness, $\aleph _0$-scatteredness with respect to character, $C$-closedness, sequentiality (the last holds under MA or $2^{\aleph _0}<2^{\aleph _1}$). Metrizable-approximable spaces are studied: every compact space in this class has a dense, Čech-complete, paracompact subspace; moreover, if $X$ is linearly ordered, then $X$ contains a dense metrizable subspace.
LA - eng
KW - $\mathcal {P}$-approximable space; Lindelöf $\Sigma $-space; compact; metrizable; $C$-closed; sequential; linearly ordered; Lindelöf -space; metrizable-approximable spaces; - approximable spaces; countable tightness; -scatteredness; compact space; dense, Čech-complete paracompact subspace
UR - http://eudml.org/doc/247317
ER -

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