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  "body": "\n## Definition\n\nLet $L$ be a [[complete lattice|D7A0ZGZT5S]].\nIt is a *frame* if for every $a \\in L$ and every family $(b_i)_{i \\in I}$ of elements of $L$,\n$$a \\wedge \\bigvee_{i \\in I} b_i = \\bigvee_{i \\in I} (a \\wedge b_i).$$\nA *frame homomorphism* between frames is a map that preserves finite meets, including the top element, and arbitrary joins, including the bottom element.\n\n## Remarks\n\nExample: for a [[topological space|0CQHFPGJ45]] $X$, the [[open sets|02BH3MCBQB]] of $X$ ordered by inclusion form a frame $\\Omega(X)$; joins are unions, finite meets are intersections, and $U \\cap \\bigcup_i V_i = \\bigcup_i (U \\cap V_i)$.\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 $(a \\wedge b) \\vee (a \\wedge c) = \\set{0}$.\nA complete lattice is a frame if and only if it is a [[Heyting algebra|NG943XNQWH]], with implication $a \\to b = \\bigvee \\set{c : a \\wedge c \\le b}$.\nEvery frame is a [[distributive lattice|0HM0HQCK6N]].\nAs ordered sets, frames, locales and complete Heyting algebras are the same objects; the three names differ in the maps one uses between them.\n",
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