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It is well known that: Theorem. For a locale (resp. topological space) $X$, the following are equivalent: $X$ is compact, i.e. every open cover of $X$ has a finite subcover. For every locale (resp. ...

Let $f : X \rightarrow Y$ be a continuous map. Suppose that, for each collection of open sets $\{ V_i \}_{i \in I}\subset X $, $$ \bigcup_{U \subset Y \text{ open}, \ f^{-1}(U) \subset \bigcup_{i \in ...