# Resolvability in c.c.c. generic extensions

## TL;DR

This paper explores the conditions under which topological spaces are resolvable in c.c.c. generic extensions, linking resolvability to monotonic functions and set-theoretic assumptions, and characterizing when spaces gain resolvability after forcing.

## Contribution

It establishes equivalences for ω₁-resolvability in c.c.c. extensions, characterizes spaces that are monotonically ω₁-resolvable, and examines the impact of set-theoretic assumptions on resolvability.

## Key findings

- ω₁-resolvability is equivalent to monotonic ω₁-resolvability in certain extensions
- Certain c.c.c. spaces with specified density and size are monotonically ω₁-resolvable
- Existence of non-monotonically ω₁-resolvable spaces under large cardinal assumptions

## Abstract

Every crowded space $X$ is ${\omega}$-resolvable in the c.c.c generic extension $V^{Fn(|X|,2})$ 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 $$\{x\in X: f(x)\ge {\alpha} \}\subset^{dense}X $$ 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 $V^{Fn({\omega_1},2)}$. We investigate which spaces are monotonically $\omega_1$-resolvable. We show that if a topological space $X$ is c.c.c, and ${\omega}_1\le \Delta(X)\le |X|<{\omega}_{\omega}$, 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 $V^{Fn({\omega},2)}$? We show that (i) if $V=L$ then every crowded c.c.c. space $X$ is ${\omega}$-resolvable in $V^{Fn({\omega},2)}$, (ii) if there is no weakly inaccesssible cardinals, then every crowded space $X$ is ${\omega}$-resolvable in $V^{Fn({\omega}_1,2)}$. On the other hand, it is also consistent that there is a crowded space $X$ with $|X|=\Delta(X)={\omega_1}$ such that $X$ remains irresolvable after adding a single Cohen real.