Resolvability in c.c.c. generic extensions

Soukup, Lajos, and Stanley, Adrienne. "Resolvability in c.c.c. generic extensions." Commentationes Mathematicae Universitatis Carolinae 58.4 (2017): 519-529. <http://eudml.org/doc/294199>.

@article{Soukup2017,
abstract = {Every crowded space $X$ is $\{\omega \}$-resolvable in the c.c.c. generic extension $V^\{\operatorname\{Fn\}(|X|,2)\}$ of the ground model. We investigate what we can say about $\{\lambda \}$-resolvability in c.c.c. generic extensions for $\lambda > \omega $. A topological space is monotonically $\omega _1$-resolvable if there is a function $f:X\rightarrow \omega _1$ such that \[ \lbrace x\in X: f(x)\ge \{\alpha \}\rbrace \subset ^\{dense\}X \] for each $\{\alpha \}< \omega _1$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is $\{\omega \}_1$-resolvable in some c.c.c. generic extension; (2) $X$ is monotonically $\omega _1$-resolvable; (3) $X$ is $\{\omega \}_1$-resolvable in the Cohen-generic extension $V^\{\operatorname\{Fn\}(\omega _1,2)\}$. We investigate which spaces are monotonically $\omega _1$-resolvable. We show that if a topological space $X$ is c.c.c., and $\{\omega \}_1\le \Delta (X)\le |X|<\{\omega \}_\{\omega \}$, where $\Delta (X) = \min \lbrace |G| : G \ne \emptyset \mbox\{ open\}\rbrace $, then $X$ is monotonically $\omega _1$-resolvable. On the other hand, it is also consistent, modulo the existence of a measurable cardinal, that there is a space $Y$ with $|Y|=\Delta (Y)=\aleph _\omega $ which is not monotonically $\omega _1$-resolvable. The characterization of $\omega _1$-resolvability in c.c.c. generic extension raises the following question: is it true that crowded spaces from the ground model are $\{\omega \}$-resolvable in $V^\{\operatorname\{Fn\}(\{\omega \},2)\}$? We show that (i) if $V=L$ then every crowded c.c.c. space $X$ is $\{\omega \}$-resolvable in $V^\{\operatorname\{Fn\}(\{\omega \},2)\}$, (ii) if there are no weakly inaccessible cardinals, then every crowded space $X$ is $\{\omega \}$-resolvable in $V^\{\operatorname\{Fn\}(\{\omega \}_1,2)\}$. Moreover, it is also consistent, modulo a measurable cardinal, that there is a crowded space $X$ with $|X|=\Delta (X)=\omega _1$ such that $X$ remains irresolvable after adding a single Cohen real.},
author = {Soukup, Lajos, Stanley, Adrienne},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {resolvable; monotonically $\omega _1$-resolvable; measurable cardinal},
language = {eng},
number = {4},
pages = {519-529},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Resolvability in c.c.c. generic extensions},
url = {http://eudml.org/doc/294199},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Soukup, Lajos
AU - Stanley, Adrienne
TI - Resolvability in c.c.c. generic extensions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 4
SP - 519
EP - 529
AB - Every crowded space $X$ is ${\omega }$-resolvable in the c.c.c. generic extension $V^{\operatorname{Fn}(|X|,2)}$ of the ground model. We investigate what we can say about ${\lambda }$-resolvability in c.c.c. generic extensions for $\lambda > \omega $. A topological space is monotonically $\omega _1$-resolvable if there is a function $f:X\rightarrow \omega _1$ such that \[ \lbrace x\in X: f(x)\ge {\alpha }\rbrace \subset ^{dense}X \] for each ${\alpha }< \omega _1$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is ${\omega }_1$-resolvable in some c.c.c. generic extension; (2) $X$ is monotonically $\omega _1$-resolvable; (3) $X$ is ${\omega }_1$-resolvable in the Cohen-generic extension $V^{\operatorname{Fn}(\omega _1,2)}$. We investigate which spaces are monotonically $\omega _1$-resolvable. We show that if a topological space $X$ is c.c.c., and ${\omega }_1\le \Delta (X)\le |X|<{\omega }_{\omega }$, where $\Delta (X) = \min \lbrace |G| : G \ne \emptyset \mbox{ open}\rbrace $, then $X$ is monotonically $\omega _1$-resolvable. On the other hand, it is also consistent, modulo the existence of a measurable cardinal, that there is a space $Y$ with $|Y|=\Delta (Y)=\aleph _\omega $ which is not monotonically $\omega _1$-resolvable. The characterization of $\omega _1$-resolvability in c.c.c. generic extension raises the following question: is it true that crowded spaces from the ground model are ${\omega }$-resolvable in $V^{\operatorname{Fn}({\omega },2)}$? We show that (i) if $V=L$ then every crowded c.c.c. space $X$ is ${\omega }$-resolvable in $V^{\operatorname{Fn}({\omega },2)}$, (ii) if there are no weakly inaccessible cardinals, then every crowded space $X$ is ${\omega }$-resolvable in $V^{\operatorname{Fn}({\omega }_1,2)}$. Moreover, it is also consistent, modulo a measurable cardinal, that there is a crowded space $X$ with $|X|=\Delta (X)=\omega _1$ such that $X$ remains irresolvable after adding a single Cohen real.
LA - eng
KW - resolvable; monotonically $\omega _1$-resolvable; measurable cardinal
UR - http://eudml.org/doc/294199
ER -