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  "body": "\n## Definition\n\nThe category of *locales* is the opposite of the category of [[frames|306PWGH7BF]] and frame homomorphisms.\nSo a *locale* $X$ is given by a frame $\\mathcal{O}(X)$, and a *locale map* $f \\colon X \\to Y$ is a frame homomorphism $f^* \\colon \\mathcal{O}(Y) \\to \\mathcal{O}(X)$.\n\n## Remarks\n\nExample: each [[topological space|0CQHFPGJ45]] $X$ gives the locale with frame $\\Omega(X)$ of [[open sets|02BH3MCBQB]].\nEach [[continuous map|0X3WGX95FE]] $f \\colon X \\to Y$ gives the locale map with $f^* = f^{-1} \\colon \\Omega(Y) \\to \\Omega(X)$, since preimages preserve unions and finite intersections.\nA point of a locale $X$ is a locale map from the one-point locale, that is, a frame homomorphism $\\mathcal{O}(X) \\to \\set{0, 1}$.\nA locale is spatial if it comes from a topological space; the functor $\\Omega$ is an equivalence between [[sober spaces|4N4DNDR22Z]] and spatial locales.\nNon-example of spatiality: a nontrivial complete Boolean algebra without atoms is a frame with no frame homomorphisms to $\\set{0, 1}$.\nAs a locale it has no points, so it is not spatial.\nLocales are the subject of [[pointless topology|CYEMGH98WN]].\n",
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