Resolvability in c.c.c. generic extensions

Abstract

Every crowded space $X$ is $ω$-resolvable in the c.c.c generic extension $V^{Fn(|X|,2})$ of the ground model. We investigate what we can say about $λ$-resolvability in c.c.c-generic extensions for $λ>ω$? A topological space is "monotonically $ω_1$-resolvable" if there is a function $f:X\to {ω_1}$ such that $$\{x\in X: f(x)\ge α \}\subset^{dense}X $$ for each $α<{ω_1}$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is $ω_1$-resolvable in some c.c.c-generic extension, (2) $X$ is monotonically $ω_1$-resolvable. (3) $X$ is $ω_1$-resolvable in the Cohen-generic extension $V^{Fn({ω_1},2)}$. We investigate which spaces are monotonically $ω_1$-resolvable. We show that if a topological space $X$ is c.c.c, and $ω_1\le Δ(X)\le |X|<ω_ω$, then $X$ is monotonically $ω_1$-resolvable. On the other hand, it is also consistent, modulo the existence of a measurable cardinal, that there is a space $Y$ with $|Y|=Δ(Y)=\aleph_ω$ which is not monotonically $ω_1$-resolvable. The characterization of ${ω_1}$-resolvability in c.c.c generic extension raises the following question: is it true that crowded spaces from the ground model are $ω$-resolvable in $V^{Fn(ω,2)}$? We show that (i) if $V=L$ then every crowded c.c.c. space $X$ is $ω$-resolvable in $V^{Fn(ω,2)}$, (ii) if there is no weakly inaccesssible cardinals, then every crowded space $X$ is $ω$-resolvable in $V^{Fn(ω_1,2)}$. On the other hand, it is also consistent that there is a crowded space $X$ with $|X|=Δ(X)={ω_1}$ such that $X$ remains irresolvable after adding a single Cohen real.