| Title:
|
Spaces of continuous functions, box products and almost-$\omega$-resolvable spaces (English) |
| Author:
|
Tamariz-Mascarúa, A. |
| Author:
|
Villegas-Rodríguez, H. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
43 |
| Issue:
|
4 |
| Year:
|
2002 |
| Pages:
|
687-705 |
| . |
| Category:
|
math |
| . |
| Summary:
|
A dense-in-itself space $X$ is called {\it $C_\square$-discrete} if the space of real continuous functions on $X$ with its box topology, $C_\square(X)$, is a discrete space. A space $X$ is called {\it almost-$\omega$-resolvable} provided that $X$ is the union of a countable increasing family of subsets each of them with an empty interior. We analyze these classes of spaces by determining their relations with $\kappa$-resolvable and almost resolvable spaces. We prove that every almost-$\omega$-resolvable space is $C_\square$-discrete, and that these classes coincide in the realm of completely regular spaces. Also, we prove that almost resolvable spaces and almost-$\omega$-resolvable spaces are two different classes of spaces if there exists a measurable cardinal. Finally, we prove that it is consistent with $ZFC$ that every dense-in-itself space is almost-$\omega$-resolvable, and that the existence of a measurable cardinal is equiconsistent with the existence of a Tychonoff space without isolated points which is not almost-$\omega$-resolvable. (English) |
| Keyword:
|
box product |
| Keyword:
|
$\kappa$-resolvable space |
| Keyword:
|
almost resolvable space |
| Keyword:
|
almost-$\omega$-resolvable space |
| Keyword:
|
Baire irresolvable space |
| Keyword:
|
measurable cardinals |
| MSC:
|
54A35 |
| MSC:
|
54B10 |
| MSC:
|
54C35 |
| MSC:
|
54F65 |
| idZBL:
|
Zbl 1090.54011 |
| idMR:
|
MR2045790 |
| . |
| Date available:
|
2009-01-08T19:26:20Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119357 |
| . |
| Reference:
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| . |
