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Keywords:
resolvable; monotonically $\omega _1$-resolvable; measurable cardinal
resolvable; monotonically $\omega _1$-resolvable; measurable cardinal
Summary:
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 $f:X\to \omega _1$ such that \begin{displaymath} \{x\in X: f(x)\geq {\alpha}\}\subset^{dense}X \end{displaymath} 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 $\Delta(X) = \min\{ |G| : G \ne \emptyset \mbox{ open}\}$, 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.
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