G δ -modification of compacta and cardinal invariants

Aleksander V. Arhangel'skii

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 1, page 95-101
  • ISSN: 0010-2628

Abstract

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Given a space X , its G δ -subsets form a basis of a new space X ω , called the G δ -modification of X . We study how the assumption that the G δ -modification X ω is homogeneous influences properties of X . If X is first countable, then X ω is discrete and, hence, homogeneous. Thus, X ω is much more often homogeneous than X itself. We prove that if X is a compact Hausdorff space of countable tightness such that the G δ -modification of X is homogeneous, then the weight w ( X ) of X does not exceed 2 ω (Theorem 1). We also establish that if a compact Hausdorff space of countable tightness is covered by a family of G δ -subspaces of the weight ≤ c = 2 ω , then the weight of X is not greater than 2 ω (Theorem 4). Several other related results are obtained, a few new open questions are formulated. Fedorchuk’s hereditarily separable compactum of the cardinality greater than c = 2 ω is shown to be G δ -homogeneous under CH. Of course, it is not homogeneous when given its own topology.

How to cite

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Arhangel'skii, Aleksander V.. "$G_\delta $-modification of compacta and cardinal invariants." Commentationes Mathematicae Universitatis Carolinae 47.1 (2006): 95-101. <http://eudml.org/doc/249857>.

@article{Arhangelskii2006,
abstract = {Given a space $X$, its $G_\delta $-subsets form a basis of a new space $X_\omega $, called the $G_\delta $-modification of $X$. We study how the assumption that the $G_\delta $-modification $X_\omega $ is homogeneous influences properties of $X$. If $X$ is first countable, then $X_\omega $ is discrete and, hence, homogeneous. Thus, $X_\omega $ is much more often homogeneous than $X$ itself. We prove that if $X$ is a compact Hausdorff space of countable tightness such that the $G_\delta $-modification of $X$ is homogeneous, then the weight $w(X)$ of $X$ does not exceed $2^\omega $ (Theorem 1). We also establish that if a compact Hausdorff space of countable tightness is covered by a family of $G_\delta $-subspaces of the weight $\le c=2^\omega $, then the weight of $X$ is not greater than $2^\omega $ (Theorem 4). Several other related results are obtained, a few new open questions are formulated. Fedorchuk’s hereditarily separable compactum of the cardinality greater than $c=2^\omega $ is shown to be $G_\delta $-homogeneous under CH. Of course, it is not homogeneous when given its own topology.},
author = {Arhangel'skii, Aleksander V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {weight; tightness; $G_\delta $-modification; character; Lindelöf degree; homogeneous space; weight; tightness; -modification; character; Lindelöf degree; homogeneous space},
language = {eng},
number = {1},
pages = {95-101},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {$G_\delta $-modification of compacta and cardinal invariants},
url = {http://eudml.org/doc/249857},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
TI - $G_\delta $-modification of compacta and cardinal invariants
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 1
SP - 95
EP - 101
AB - Given a space $X$, its $G_\delta $-subsets form a basis of a new space $X_\omega $, called the $G_\delta $-modification of $X$. We study how the assumption that the $G_\delta $-modification $X_\omega $ is homogeneous influences properties of $X$. If $X$ is first countable, then $X_\omega $ is discrete and, hence, homogeneous. Thus, $X_\omega $ is much more often homogeneous than $X$ itself. We prove that if $X$ is a compact Hausdorff space of countable tightness such that the $G_\delta $-modification of $X$ is homogeneous, then the weight $w(X)$ of $X$ does not exceed $2^\omega $ (Theorem 1). We also establish that if a compact Hausdorff space of countable tightness is covered by a family of $G_\delta $-subspaces of the weight $\le c=2^\omega $, then the weight of $X$ is not greater than $2^\omega $ (Theorem 4). Several other related results are obtained, a few new open questions are formulated. Fedorchuk’s hereditarily separable compactum of the cardinality greater than $c=2^\omega $ is shown to be $G_\delta $-homogeneous under CH. Of course, it is not homogeneous when given its own topology.
LA - eng
KW - weight; tightness; $G_\delta $-modification; character; Lindelöf degree; homogeneous space; weight; tightness; -modification; character; Lindelöf degree; homogeneous space
UR - http://eudml.org/doc/249857
ER -

References

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