Extraresolvability and cardinal arithmetic

Alas, Ofelia Teresa, García-Ferreira, Salvador, and Tomita, Artur Hideyuki. "Extraresolvability and cardinal arithmetic." Commentationes Mathematicae Universitatis Carolinae 40.2 (1999): 279-292. <http://eudml.org/doc/248383>.

@article{Alas1999,
abstract = {Following Malykhin, we say that a space $X$ is extraresolvable if $X$ contains a family $\mathcal \{D\}$ of dense subsets such that $|\mathcal \{D\}| > \Delta (X)$ and the intersection of every two elements of $\mathcal \{D\}$ is nowhere dense, where $\Delta (X) = \min \lbrace |U|: U$ is a nonempty open subset of $X\rbrace $ is the dispersion character of $X$. We show that, for every cardinal $\kappa $, there is a compact extraresolvable space of size and dispersion character $2^\kappa $. In connection with some cardinal inequalities, we prove the equivalence of the following statements: 1) $2^\kappa < 2^\{\{\kappa \}^\{+\}\}$, 2) $(\kappa ^\{+\})^\{\kappa \}$ is extraresolvable and 3) $A(\kappa ^\{+\})^\{\kappa \}$ is extraresolvable, where $A(\kappa ^\{+\})$ is the one-point compactification of the discrete space $\kappa ^\{+\}$. For a regular cardinal $\kappa \ge \omega $, we show that the following are equivalent: 1) $2^\{< \kappa \} < 2^\{\kappa \}$; 2) $G(\kappa ,\kappa )$ is extraresolvable; 3) $G(\kappa ,\kappa )^\lambda $ is extraresolvable for all $\lambda < \kappa $; and 4) there exists a space $X$ such that $X^\lambda $ is extraresolvable, for all $\lambda < \kappa $, and $X^\kappa $ is not extraresolvable, where $G(\kappa ,\kappa )= \lbrace x \in \lbrace 0,1\rbrace ^\kappa : |\lbrace \xi < \kappa : x_\xi \ne 0 \rbrace | < \kappa \rbrace $ for every $\kappa \ge \omega $. It is also shown that if $X$ is extraresolvable and $\Delta (X) = |X|$, then all powers of $X$ have a dense extraresolvable subset, and $\lambda ^\kappa $ contains a dense extraresolvable subspace for every cardinal $\lambda \ge 2$ and for every infinite cardinal $\kappa $. For an infinite cardinal $\kappa $, if $2^\kappa > \{\mathfrak \{c\}\}$, then there is a totally bounded, connected, extraresolvable, topological Abelian group of size and dispersion character equal to $\kappa $, and if $\kappa = \kappa ^\omega $, then there is an $\omega $-bounded, normal, connected, extraresolvable, topological Abelian group of size and dispersion character equal to $\kappa $.},
author = {Alas, Ofelia Teresa, García-Ferreira, Salvador, Tomita, Artur Hideyuki},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {extraresolvable; $\kappa $-resolvable; extraresolvability; product; topological group},
language = {eng},
number = {2},
pages = {279-292},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Extraresolvability and cardinal arithmetic},
url = {http://eudml.org/doc/248383},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Alas, Ofelia Teresa
AU - García-Ferreira, Salvador
AU - Tomita, Artur Hideyuki
TI - Extraresolvability and cardinal arithmetic
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 2
SP - 279
EP - 292
AB - Following Malykhin, we say that a space $X$ is extraresolvable if $X$ contains a family $\mathcal {D}$ of dense subsets such that $|\mathcal {D}| > \Delta (X)$ and the intersection of every two elements of $\mathcal {D}$ is nowhere dense, where $\Delta (X) = \min \lbrace |U|: U$ is a nonempty open subset of $X\rbrace $ is the dispersion character of $X$. We show that, for every cardinal $\kappa $, there is a compact extraresolvable space of size and dispersion character $2^\kappa $. In connection with some cardinal inequalities, we prove the equivalence of the following statements: 1) $2^\kappa < 2^{{\kappa }^{+}}$, 2) $(\kappa ^{+})^{\kappa }$ is extraresolvable and 3) $A(\kappa ^{+})^{\kappa }$ is extraresolvable, where $A(\kappa ^{+})$ is the one-point compactification of the discrete space $\kappa ^{+}$. For a regular cardinal $\kappa \ge \omega $, we show that the following are equivalent: 1) $2^{< \kappa } < 2^{\kappa }$; 2) $G(\kappa ,\kappa )$ is extraresolvable; 3) $G(\kappa ,\kappa )^\lambda $ is extraresolvable for all $\lambda < \kappa $; and 4) there exists a space $X$ such that $X^\lambda $ is extraresolvable, for all $\lambda < \kappa $, and $X^\kappa $ is not extraresolvable, where $G(\kappa ,\kappa )= \lbrace x \in \lbrace 0,1\rbrace ^\kappa : |\lbrace \xi < \kappa : x_\xi \ne 0 \rbrace | < \kappa \rbrace $ for every $\kappa \ge \omega $. It is also shown that if $X$ is extraresolvable and $\Delta (X) = |X|$, then all powers of $X$ have a dense extraresolvable subset, and $\lambda ^\kappa $ contains a dense extraresolvable subspace for every cardinal $\lambda \ge 2$ and for every infinite cardinal $\kappa $. For an infinite cardinal $\kappa $, if $2^\kappa > {\mathfrak {c}}$, then there is a totally bounded, connected, extraresolvable, topological Abelian group of size and dispersion character equal to $\kappa $, and if $\kappa = \kappa ^\omega $, then there is an $\omega $-bounded, normal, connected, extraresolvable, topological Abelian group of size and dispersion character equal to $\kappa $.
LA - eng
KW - extraresolvable; $\kappa $-resolvable; extraresolvability; product; topological group
UR - http://eudml.org/doc/248383
ER -