# Topological games and product spaces

• Volume: 43, Issue: 4, page 675-685
• ISSN: 0010-2628

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## Abstract

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In this paper, we deal with the product of spaces which are either $𝒢$-spaces or ${𝒢}_{p}$-spaces, for some $p\in {\omega }^{*}$. These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are $𝒢$-spaces, and every ${𝒢}_{p}$-space is a $𝒢$-space, for every $p\in {\omega }^{*}$. We prove that if $\left\{{X}_{\mu }:\mu <{\omega }_{1}\right\}$ is a set of spaces whose product $X={\prod }_{\mu <{\omega }_{1}}{X}_{\mu }$ is a $𝒢$-space, then there is $A\in {\left[{\omega }_{1}\right]}^{\le \omega }$ such that ${X}_{\mu }$ is countably compact for every $\mu \in {\omega }_{1}\setminus A$. As a consequence, ${X}^{{\omega }_{1}}$ is a $𝒢$-space iff ${X}^{{\omega }_{1}}$ is countably compact, and if ${X}^{{2}^{𝔠}}$ is a $𝒢$-space, then all powers of $X$ are countably compact. It is easy to prove that the product of a countable family of ${𝒢}_{p}$ spaces is a ${𝒢}_{p}$-space, for every $p\in {\omega }^{*}$. For every $1\le n<\omega$, we construct a space $X$ such that ${X}^{n}$ is countably compact and ${X}^{n+1}$ is not a $𝒢$-space. If $p,q\in {\omega }^{*}$ are $RK$-incomparable, then we construct a ${𝒢}_{p}$-space $X$ and a ${𝒢}_{q}$-space $Y$ such that $X×Y$ is not a $𝒢$-space. We give an example of two free ultrafilters $p$ and $q$ on $\omega$ such that $p{<}_{RK}q$, $p$ and $q$ are $RF$-incomparable, $p{\approx }_{C}q$ (${\le }_{C}$ is the Comfort order on ${\omega }^{*}$) and there are a ${𝒢}_{p}$-space $X$ and a ${𝒢}_{q}$-space $Y$ whose product $X×Y$ is not a $𝒢$-space.

## How to cite

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García-Ferreira, Salvador, González-Silva, R. A., and Tomita, Artur Hideyuki. "Topological games and product spaces." Commentationes Mathematicae Universitatis Carolinae 43.4 (2002): 675-685. <http://eudml.org/doc/248973>.

@article{García2002,
abstract = {In this paper, we deal with the product of spaces which are either $\mathcal \{G\}$-spaces or $\mathcal \{G\}_p$-spaces, for some $p \in \omega ^*$. These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are $\{\mathcal \{G\}\}$-spaces, and every $\mathcal \{G\}_p$-space is a $\mathcal \{G\}$-space, for every $p \in \omega ^*$. We prove that if $\lbrace X_\mu : \mu < \omega _1 \rbrace$ is a set of spaces whose product $X= \prod _\{\mu < \omega _1\}X_ \mu$ is a $\mathcal \{G\}$-space, then there is $A \in [\omega _1]^\{\le \omega \}$ such that $X_\mu$ is countably compact for every $\mu \in \omega _1 \setminus A$. As a consequence, $X^\{\omega _1\}$ is a $\mathcal \{G\}$-space iff $X^\{\omega _1\}$ is countably compact, and if $X^\{2^\{\mathfrak \{c\}\}\}$ is a $\mathcal \{G\}$-space, then all powers of $X$ are countably compact. It is easy to prove that the product of a countable family of $\mathcal \{G\}_p$ spaces is a $\mathcal \{G\}_p$-space, for every $p \in \omega ^*$. For every $1 \le n < \omega$, we construct a space $X$ such that $X^n$ is countably compact and $X^\{n+1\}$ is not a $\mathcal \{G\}$-space. If $p, q \in \omega ^*$ are $RK$-incomparable, then we construct a $\mathcal \{G\}_p$-space $X$ and a $\mathcal \{G\}_q$-space $Y$ such that $X \times Y$ is not a $\mathcal \{G\}$-space. We give an example of two free ultrafilters $p$ and $q$ on $\omega$ such that $p <_\{RK\} q$, $p$ and $q$ are $RF$-incomparable, $p \approx _C q$ ($\le _C$ is the Comfort order on $\omega ^*$) and there are a $\mathcal \{G\}_p$-space $X$ and a $\mathcal \{G\}_q$-space $Y$ whose product $X \times Y$ is not a $\mathcal \{G\}$-space.},
author = {García-Ferreira, Salvador, González-Silva, R. A., Tomita, Artur Hideyuki},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$RF$-order; $RK$-order; Comfort-order; $p$-limit; $p$-compact; $\mathcal \{G\}$-space; $\mathcal \{G\}_p$-space; countably compact; product of spaces; -spaces; ultrafilters},
language = {eng},
number = {4},
pages = {675-685},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Topological games and product spaces},
url = {http://eudml.org/doc/248973},
volume = {43},
year = {2002},
}

TY - JOUR
AU - González-Silva, R. A.
AU - Tomita, Artur Hideyuki
TI - Topological games and product spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 4
SP - 675
EP - 685
AB - In this paper, we deal with the product of spaces which are either $\mathcal {G}$-spaces or $\mathcal {G}_p$-spaces, for some $p \in \omega ^*$. These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are ${\mathcal {G}}$-spaces, and every $\mathcal {G}_p$-space is a $\mathcal {G}$-space, for every $p \in \omega ^*$. We prove that if $\lbrace X_\mu : \mu < \omega _1 \rbrace$ is a set of spaces whose product $X= \prod _{\mu < \omega _1}X_ \mu$ is a $\mathcal {G}$-space, then there is $A \in [\omega _1]^{\le \omega }$ such that $X_\mu$ is countably compact for every $\mu \in \omega _1 \setminus A$. As a consequence, $X^{\omega _1}$ is a $\mathcal {G}$-space iff $X^{\omega _1}$ is countably compact, and if $X^{2^{\mathfrak {c}}}$ is a $\mathcal {G}$-space, then all powers of $X$ are countably compact. It is easy to prove that the product of a countable family of $\mathcal {G}_p$ spaces is a $\mathcal {G}_p$-space, for every $p \in \omega ^*$. For every $1 \le n < \omega$, we construct a space $X$ such that $X^n$ is countably compact and $X^{n+1}$ is not a $\mathcal {G}$-space. If $p, q \in \omega ^*$ are $RK$-incomparable, then we construct a $\mathcal {G}_p$-space $X$ and a $\mathcal {G}_q$-space $Y$ such that $X \times Y$ is not a $\mathcal {G}$-space. We give an example of two free ultrafilters $p$ and $q$ on $\omega$ such that $p <_{RK} q$, $p$ and $q$ are $RF$-incomparable, $p \approx _C q$ ($\le _C$ is the Comfort order on $\omega ^*$) and there are a $\mathcal {G}_p$-space $X$ and a $\mathcal {G}_q$-space $Y$ whose product $X \times Y$ is not a $\mathcal {G}$-space.
LA - eng
KW - $RF$-order; $RK$-order; Comfort-order; $p$-limit; $p$-compact; $\mathcal {G}$-space; $\mathcal {G}_p$-space; countably compact; product of spaces; -spaces; ultrafilters
UR - http://eudml.org/doc/248973
ER -

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