# Article

 Title: Topological games and product spaces (English) Author: García-Ferreira, S. Author: González-Silva, R. A. Author: Tomita, A. H. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 43 Issue: 4 Year: 2002 Pages: 675-685 . Category: math . Summary: In this paper, we deal with the product of spaces which are either $\Cal G$-spaces or $\Cal G_p$-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 ${\Cal G}$-spaces, and every $\Cal G_p$-space is a $\Cal G$-space, for every $p \in \omega^*$. We prove that if $\{ X_\mu : \mu < \omega_1 \}$ is a set of spaces whose product $X= \prod_{\mu < \omega_1}X_ \mu$ is a $\Cal G$-space, then there is $A \in [\omega_1]^{\leq \omega}$ such that $X_\mu$ is countably compact for every $\mu \in \omega_1 \setminus A$. As a consequence, $X^{\omega_1}$ is a $\Cal G$-space iff $X^{\omega_1}$ is countably compact, and if $X^{2^{\frak c}}$ is a $\Cal G$-space, then all powers of $X$ are countably compact. It is easy to prove that the product of a countable family of $\Cal G_p$ spaces is a $\Cal G_p$-space, for every $p \in \omega^*$. For every $1 \leq n < \omega$, we construct a space $X$ such that $X^n$ is countably compact and $X^{n+1}$ is not a $\Cal G$-space. If $p, q \in \omega^*$ are $RK$-incomparable, then we construct a $\Cal G_p$-space $X$ and a $\Cal G_q$-space $Y$ such that $X \times Y$ is not a $\Cal G$-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, $p \approx_C q$ ($\leq_C$ is the {\it Comfort} order on $\omega^*$) and there are a $\Cal G_p$-space $X$ and a $\Cal G_q$-space $Y$ whose product $X \times Y$ is not a $\Cal G$-space. (English) Keyword: $RF$-order Keyword: $RK$-order Keyword: {\it Comfort}-order Keyword: $p$-limit Keyword: $p$-compact Keyword: $\Cal G$-space Keyword: $\Cal G_p$-space Keyword: countably compact MSC: 03E05 MSC: 03E35 MSC: 54A25 MSC: 54A35 MSC: 54B10 MSC: 54D99 MSC: 91A44 idZBL: Zbl 1090.54005 idMR: MR2045789 . Date available: 2009-01-08T19:26:12Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119356 . Reference: [Be] Bernstein A.R.: A new kind of compactness for topological spaces.Fund. Math. 66 (1970), 185-193. Zbl 0198.55401, MR 0251697 Reference: [Boo] Booth D.: Ultrafilters on a countable set.Ann. Math. Logic 2 (1970), 1-24. Zbl 0231.02067, MR 0277371 Reference: [Bo] Bouziad A.: The Ellis theorem and continuity in groups.Topology Appl. 50 (1993), 73-80. Zbl 0827.54018, MR 1217698 Reference: [CN] Comfort W., Negrepontis S.: The Theory of Ultrafilters.Springer-Verlag, Berlin, 1974. Zbl 0298.02004, MR 0396267 Reference: [En] Engelking R.: General Topology.Sigma Series in Pure Mathematics Vol. 6, Heldermann Verlag Berlin, 1989. Zbl 0684.54001, MR 1039321 Reference: [Fro] Frolík Z.: Sums of ultrafilters.Bull. Amer. Math. Soc. 73 (1967), 87-91. MR 0203676 Reference: [G] García-Ferreira S.: Three orderings on $ømega^*$.Topology Appl. 50 (1993), 199-216. MR 1227550 Reference: [GG] García-Ferreira S., González-Silva R.A.: Topological games defined by ultrafilters.to appear in Topology Appl. MR 2057882 Reference: [GS] Ginsburg J., Saks V.: Some applications of ultrafilters in topology.Pacific J. Math. 57 (1975), 403-418. Zbl 0288.54020, MR 0380736 Reference: [Gru] Gruenhage G.: Infinite games and generalizations of first countable spaces,.Topology Appl. 6 (1976), 339-352. Zbl 0327.54019, MR 0413049 Reference: [HST] Hrušák M., Sanchis M., Tamariz-Mascarúa A.: Ultrafilters, special functions and pseudocompactness.in process. Reference: [Ku] Kunen K.: Weak $P$-points in $N^*$.Colloq. Math. Soc. János Bolyai 23, Topology, Budapest (Hungary), pp.741-749. Zbl 0435.54021, MR 0588822 Reference: [Si] Simon P.: Applications of independent linked families.Topology, Theory and Applications (Eger, 1983), Colloq. Math. Soc. János Bolyai 41 (1985), 561-580. Zbl 0615.54004, MR 0863940 Reference: [Va] 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. MR 0776631 .

## Files

Files Size Format View
CommentatMathUnivCarolRetro_43-2002-4_8.pdf 248.6Kb application/pdf View/Open

Partner of