Coloring Cantor sets and resolvability of pseudocompact spaces

Juhász, István, Soukup, Lajos, and Szentmiklóssy, Zoltán. "Coloring Cantor sets and resolvability of pseudocompact spaces." Commentationes Mathematicae Universitatis Carolinae 59.4 (2018): 523-529. <http://eudml.org/doc/294347>.

@article{Juhász2018,
abstract = {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].},
author = {Juhász, István, Soukup, Lajos, Szentmiklóssy, Zoltán},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {pseudocompact; feebly compact; resolvable; Baire space; coloring; Cantor set},
language = {eng},
number = {4},
pages = {523-529},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Coloring Cantor sets and resolvability of pseudocompact spaces},
url = {http://eudml.org/doc/294347},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Juhász, István
AU - Soukup, Lajos
AU - Szentmiklóssy, Zoltán
TI - Coloring Cantor sets and resolvability of pseudocompact spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 4
SP - 523
EP - 529
AB - 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].
LA - eng
KW - pseudocompact; feebly compact; resolvable; Baire space; coloring; Cantor set
UR - http://eudml.org/doc/294347
ER -