Lemma C.4

For every at most countable index type $I$ in universe $u$, and every family of topological spaces $X_i$ whose underlying types lie in universe $u$, suppose that for each $i\in I$ there exist an at most countable index type $J_i$ in universe $u$, compact Hausdorff spaces $K_{i,j}$ in universe $u$, and continuous maps $f_{i,j}:K_{i,j}\to X_i$ whose images cover $X_i$, such that a subset $A\subseteq X_i$ is open if and only if every $f_{i,j}^{-1}(A)$ is open. Then the topological disjoint union $S=\coprod_{i\in I}X_i$ also admits an at most countable index type $L$ in universe $u$, compact Hausdorff spaces $H_\ell$ in universe $u$, and continuous maps $h_\ell:H_\ell\to S$ whose images cover $S$, such that a subset $B\subseteq S$ is open if and only if every $h_\ell^{-1}(B)$ is open. All these countable index types may be finite or empty, and the spaces may be empty; in particular, $I=\varnothing$ is included.