| Title:
|
Martin's Axiom and $\omega$-resolvability of Baire spaces (English) |
| Author:
|
Casarrubias-Segura, Fidel |
| Author:
|
Hernández-Hernández, Fernando |
| Author:
|
Tamariz-Mascarúa, Ángel |
| Language:
|
English |
| Journal:
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Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
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1213-7243 (online) |
| Volume:
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51 |
| Issue:
|
3 |
| Year:
|
2010 |
| Pages:
|
519-540 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We prove that, assuming MA, every crowded $T_0$ space $X$ is $\omega$-resolvable if it satisfies one of the following properties: (1) it contains a $\pi$-network of cardinality $< \frak{c}$ constituted by infinite sets, (2) $\chi(X) < \frak{c}$, (3) $X$ is a $T_2$ Baire space and $c(X) \leq \aleph_0$ and (4) $X$ is a $T_1$ Baire space and has a network $\Cal{N}$ with cardinality $< \frak{c}$ and such that the collection of the finite elements in it constitutes a $\sigma$-locally finite family. Furthermore, we prove that the existence of a $T_1$ Baire irresolvable space is equivalent to the existence of a $T_1$ Baire $\omega$-irresolvable space, and each of these statements is equivalent to the existence of a $T_1$ almost-$\omega$-irresolvable space. Finally, we prove that the minimum cardinality of a $\pi$-network with infinite elements of a space $\operatorname{Seq}(u_t)$ is strictly greater than $\aleph_0$. (English) |
| Keyword:
|
Martin's Axiom |
| Keyword:
|
Baire spaces |
| Keyword:
|
resolvable spaces |
| Keyword:
|
$\omega$-resolvable spaces |
| Keyword:
|
almost resolvable spaces |
| Keyword:
|
almost-$\omega$-resolvable spaces |
| Keyword:
|
infinite $\pi$-network |
| MSC:
|
54A10 |
| MSC:
|
54A35 |
| MSC:
|
54D10 |
| MSC:
|
54E52 |
| idZBL:
|
Zbl 1224.54068 |
| idMR:
|
MR2741885 |
| . |
| Date available:
|
2010-09-02T14:23:08Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140728 |
| . |
| Reference:
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