| 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 |
| . |
