Lemma C.3

For any topological spaces $X,Y$ whose underlying types lie in universe $u$, suppose there exist an at most countable index type $I$ in universe $u$, compact Hausdorff spaces $K_i$ in universe $u$, and continuous maps $f_i:K_i\to X$ whose images cover $X$, such that a subset $A\subseteq X$ is open if and only if every $f_i^{-1}(A)$ is open. For every function $q:X\to Y$ that is surjective and satisfies, for every $B\subseteq Y$, that $B$ is open if and only if $q^{-1}(B)$ is open, there exist an at most countable index type $J$ in universe $u$, compact Hausdorff spaces $L_j$ in universe $u$, and continuous maps $h_j:L_j\to Y$ whose images cover $Y$, such that $B\subseteq Y$ is open if and only if every $h_j^{-1}(B)$ is open. The index types and compact Hausdorff spaces may be empty; if $X$ is empty, surjectivity forces $Y$ to be empty as well.