# Topological games and product spaces

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

top Access to full text Full (PDF) Access to full text

## Abstract

top
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.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.