# Article

Full entry | PDF   (0.2 MB)
Keywords:
$RF$-order; $RK$-order; {\it Comfort}-order; $p$-limit; $p$-compact; $\Cal G$-space; $\Cal G_p$-space; countably compact
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.
References:
[Be] Bernstein A.R.: A new kind of compactness for topological spaces. Fund. Math. 66 (1970), 185-193. MR 0251697 | Zbl 0198.55401
[Boo] Booth D.: Ultrafilters on a countable set. Ann. Math. Logic 2 (1970), 1-24. MR 0277371 | Zbl 0231.02067
[Bo] Bouziad A.: The Ellis theorem and continuity in groups. Topology Appl. 50 (1993), 73-80. MR 1217698 | Zbl 0827.54018
[CN] Comfort W., Negrepontis S.: The Theory of Ultrafilters. Springer-Verlag, Berlin, 1974. MR 0396267 | Zbl 0298.02004
[En] Engelking R.: General Topology. Sigma Series in Pure Mathematics Vol. 6, Heldermann Verlag Berlin, 1989. MR 1039321 | Zbl 0684.54001
[Fro] Frolík Z.: Sums of ultrafilters. Bull. Amer. Math. Soc. 73 (1967), 87-91. MR 0203676
[G] García-Ferreira S.: Three orderings on $ømega^*$. Topology Appl. 50 (1993), 199-216. MR 1227550
[GG] García-Ferreira S., González-Silva R.A.: Topological games defined by ultrafilters. to appear in Topology Appl. MR 2057882
[GS] Ginsburg J., Saks V.: Some applications of ultrafilters in topology. Pacific J. Math. 57 (1975), 403-418. MR 0380736 | Zbl 0288.54020
[Gru] Gruenhage G.: Infinite games and generalizations of first countable spaces. Topology Appl. 6 (1976), 339-352. MR 0413049 | Zbl 0327.54019
[HST] Hrušák M., Sanchis M., Tamariz-Mascarúa A.: Ultrafilters, special functions and pseudocompactness. in process.
[Ku] Kunen K.: Weak $P$-points in $N^*$. Colloq. Math. Soc. János Bolyai 23, Topology, Budapest (Hungary), pp.741-749. MR 0588822 | Zbl 0435.54021
[Si] Simon P.: Applications of independent linked families. Topology, Theory and Applications (Eger, 1983), Colloq. Math. Soc. János Bolyai 41 (1985), 561-580. MR 0863940 | Zbl 0615.54004
[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

Partner of