Locale

Remarks

Example: each topological space $X$ gives the locale with frame $\Omega(X)$ of open sets. Each continuous map $f \colon X \to Y$ gives the locale map with $f^* = f^{-1} \colon \Omega(Y) \to \Omega(X)$, since preimages preserve unions and finite intersections. A point of a locale $X$ is a locale map from the one-point locale, that is, a frame homomorphism $\mathcal{O}(X) \to \set{0, 1}$. A locale is spatial if it comes from a topological space; the functor $\Omega$ is an equivalence between sober spaces and spatial locales. Non-example of spatiality: a nontrivial complete Boolean algebra without atoms is a frame with no frame homomorphisms to $\set{0, 1}$. As a locale it has no points, so it is not spatial. Locales are the subject of pointless topology.