Let $\Omega$ be a set and let $\mathcal{C} \subset \mathcal{P}(\Omega)$ be some collection of subsets that covers $\Omega$, so $\bigcup_{C \in \mathcal{C}} = \Omega$. We would like to find a partition $\mathcal{P}$ of $\Omega$ that
- refines $\mathcal{C}$, so for all $P \in \mathcal{P}$ there is a $C \in \mathcal{C}$ with $P \subset C$.
- does not have greater cardinality than $\mathcal{C}$, which we take to mean that there must be a surjective function $e \colon \mathcal{C} \to \mathcal{P}$.
Note that this is definitely possible when $\mathcal{C}$ is well-orderable. Supply $\mathcal{C}$ with a well-order $\le_w$. For $C \in \mathcal{C}$, define $$ P_C = C \setminus \bigcup_{E <_w C} E. $$ If $C_1 <_w C_2$ then $P_{C_1}$ and $P_{C_2}$ will be disjoint, and for any $x \in \Omega$ there is a $\le_w$-smallest $C_0$ containing $x$, so $x \in P_{C_0}$. The function $C \mapsto P_C$ is surjective by construction.
This construction relies fundamentally on the well-orderability of $\mathcal{C}$ (or more precisely that $\mathcal{C}$ is linearly orderable such that the set $\{ C \in \mathcal{C} : x \in C \}$ has a smallest element for all $x \in \Omega$), so I'm wondering wether it is not provable in the absence of any sort of axiom of choice. Are there other properties we can impose on $\mathcal{C}$ other than well-orderability that guarantee we can always find such a $\mathcal{P}$?