# Topological games and product spaces

Salvador García-Ferreira; R. A. González-Silva; Artur Hideyuki Tomita

Commentationes Mathematicae Universitatis Carolinae (2002)

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

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topGarcía-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 - García-Ferreira, Salvador

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 -

## References

top- Bernstein A.R., A new kind of compactness for topological spaces, Fund. Math. 66 (1970), 185-193. (1970) Zbl0198.55401MR0251697
- Booth D., Ultrafilters on a countable set, Ann. Math. Logic 2 (1970), 1-24. (1970) Zbl0231.02067MR0277371
- Bouziad A., The Ellis theorem and continuity in groups, Topology Appl. 50 (1993), 73-80. (1993) Zbl0827.54018MR1217698
- Comfort W., Negrepontis S., The Theory of Ultrafilters, Springer-Verlag, Berlin, 1974. Zbl0298.02004MR0396267
- Engelking R., General Topology, Sigma Series in Pure Mathematics Vol. 6, Heldermann Verlag Berlin, 1989. Zbl0684.54001MR1039321
- Frolík Z., Sums of ultrafilters, Bull. Amer. Math. Soc. 73 (1967), 87-91. (1967) MR0203676
- García-Ferreira S., Three orderings on $\xf8meg{a}^{*}$, Topology Appl. 50 (1993), 199-216. (1993) MR1227550
- García-Ferreira S., González-Silva R.A., Topological games defined by ultrafilters, to appear in Topology Appl. MR2057882
- Ginsburg J., Saks V., Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), 403-418. (1975) Zbl0288.54020MR0380736
- Gruenhage G., Infinite games and generalizations of first countable spaces,, Topology Appl. 6 (1976), 339-352. (1976) Zbl0327.54019MR0413049
- Hrušák M., Sanchis M., Tamariz-Mascarúa A., Ultrafilters, special functions and pseudocompactness, in process.
- Kunen K., Weak $P$-points in ${N}^{*}$, Colloq. Math. Soc. János Bolyai 23, Topology, Budapest (Hungary), pp.741-749. Zbl0435.54021MR0588822
- Simon P., Applications of independent linked families, Topology, Theory and Applications (Eger, 1983), Colloq. Math. Soc. János Bolyai 41 (1985), 561-580. Zbl0615.54004MR0863940
- Vaughan J.E., Countably compact sequentially compact spaces, in: Handbook of Set-Theoretic Topology, editors J. van Mill and J. Vaughan, North-Holland, pp.571-600. MR0776631

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