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 \[ \{x\in X:f(x)\geq \alpha\} \subset^{dense}X \] 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 \(\omega_1\leq\Delta(X)\leq |X|<\omega_{\omega}\), where \(\Delta(X) = \min\{ |G|:G\neq\emptyset \text{ open}\}\), 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.