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  "body": "\n## Definition\n\nA *frame* is a complete lattice $L$ such that $a \\wedge \\bigvee_{i \\in I} b_i = \\bigvee_{i \\in I} (a \\wedge b_i)$ for every $a \\in L$ and every family $(b_i)_{i \\in I}$ in $L$.\nA *frame homomorphism* is a map between frames that preserves finite meets, including the top element, and arbitrary joins, including the bottom element.\nThe category of *locales* is the opposite of the category of frames and frame homomorphisms.\n*Pointless topology* is the study of locales.\nEach topological space $X$ gives the locale $\\Omega(X)$, the frame of [[open sets|02BH3MCBQB]] of $X$ ordered by inclusion.\nEach [[continuous map|0X3WGX95FE]] $f \\colon X \\to Y$ gives the frame homomorphism $f^{-1} \\colon \\Omega(Y) \\to \\Omega(X)$, hence a locale map $\\Omega(X) \\to \\Omega(Y)$.\n\n## Remarks\n\nExample: for $X = \\R$, the open sets form a frame, because $U \\cap \\bigcup_i V_i = \\bigcup_i (U \\cap V_i)$ for open sets $U, V_i$.\nNon-example: the lattice of subspaces of $\\R^2$ is complete but not a frame; for three distinct lines $a, b, c$ through $0$, $a \\wedge (b \\vee c) = a$ while $(a \\wedge b) \\vee (a \\wedge c) = 0$.\nA point of a locale $L$ is a frame homomorphism $L \\to \\set{0, 1}$; a locale is spatial if it is isomorphic to $\\Omega(X)$ for some space $X$.\nThe functor $\\Omega$ restricts to an equivalence between [[sober spaces|4N4DNDR22Z]] and spatial locales.\nEvery frame is a complete [[Heyting algebra|NG943XNQWH]]; Mathlib calls frames `Order.Frame`.\n",
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