| Title:
|
Coloring Cantor sets and resolvability of pseudocompact spaces (English) |
| Author:
|
Juhász, István |
| Author:
|
Soukup, Lajos |
| Author:
|
Szentmiklóssy, Zoltán |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
59 |
| Issue:
|
4 |
| Year:
|
2018 |
| Pages:
|
523-529 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(\lambda)$ picks up all the $\mu$ colors. We call a space $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and only if it is feebly compact. The main result of this paper is the following: Let $X$ be a crowded feebly compact $\pi$-regular space and $\mu$ be a fixed (finite or infinite) cardinal. If $\Phi(\lambda,\mu)$ holds for all $\lambda < \hat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. $X$ contains $\mu$ pairwise disjoint dense subsets. (Here $\hat{c}(X)$ is the smallest cardinal $\kappa$ such that $X$ does not contain $\kappa$ many pairwise disjoint open sets.) This significantly improves earlier results of [van Mill J., {Every crowded pseudocompact ccc space is resolvable}, Topology Appl. 213 (2016), 127--134], or [Ortiz-Castillo Y. F., Tomita A. H., {Crowded pseudocompact Tychonoff spaces of cellularity at most the continuum are resolvable}, Conf. talk at Toposym 2016]. (English) |
| Keyword:
|
pseudocompact |
| Keyword:
|
feebly compact |
| Keyword:
|
resolvable |
| Keyword:
|
Baire space |
| Keyword:
|
coloring |
| Keyword:
|
Cantor set |
| MSC:
|
54A25 |
| MSC:
|
54A35 |
| MSC:
|
54D30 |
| MSC:
|
54E35 |
| idZBL:
|
Zbl 06997368 |
| idMR:
|
MR3914718 |
| DOI:
|
10.14712/1213-7243.2015.261 |
| . |
| Date available:
|
2018-12-28T15:15:23Z |
| Last updated:
|
2021-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147556 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
[7] Ortiz-Castillo Y. F., Tomita A. H.: Crowded pseudocompact Tychonoff spaces of cellularity at most the continuum are resolvable.Conf. talk at Toposym 2016 available at http://www.toposym.cz//slides-Ortiz\_Castillo-2435.pdf. |
| Reference:
|
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| Reference:
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| . |
