in $O$. If $C \subseteq X$ is a regular closed set, then $C = \operatorname{cl}(f^{-1}(f^{\#}(\operatorname{int}(C))))$, and $f[C]= \operatorname{cl}(f^{\#}(\operatorname{int}(C)))$, so in particular the image of a regular closed set is regular. The preimage of any point by $f$ is either an isolated point or has empty interior. If $C, C'$ are regular closed and $f[C] = f[C']$ then $C = C'$; if $\operatorname{int}(C \cap C') = \emptyset$ then $\operatorname{int}(f[C] \cap f[C']) = \emptyset$.

\[def:regular quasi-partition\] A covering $ \mathcal C $ of a topological space is a *regular quasi-partition* if the elements of $ \mathcal C $ are nonempty, regular closed sets and $\forall A, B\in \mathcal C \ (A\ne B\Rightarrow A\cap B\subseteq\partial (A)\cap\partial (B))$.

\[quasi-part\] If $X, Y$ are compact metrizable spaces and $f:X\to Y$ is an irreducible closed surjective map, then the image $f \mathcal C = \setnew{f[C] }{C \in \mathcal C}$ of a regular quasi-partition $\mathcal C$ of $X$ is a regular quasi-partition of $Y$, and the map $C \mapsto f[C]$ is a bijection between $\mathcal C$ and $f \mathcal C$.

The fact that $C\mapsto f[C]$ is a bijection is one of the basic properties of irreducible closed surjective maps between compact metrizable spaces. The same for the 