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  "body": "\n## Summary\n\nA completely distributive lattice is a complete lattice in which arbitrary meets distribute over arbitrary joins, in the strongest possible form.\nIt is the infinite version of a [[distributive lattice|0HM0HQCK6N]].\nPower sets and the closed interval $[0, 1]$ are completely distributive, and every completely distributive lattice is a [[frame|306PWGH7BF]].\n\n## Explanations\n\n### L1 human\n\nIn a distributive lattice, one can expand $a \\wedge (b \\vee c)$ into $(a \\wedge b) \\vee (a \\wedge c)$, like multiplying out brackets.\nComplete distributivity allows this expansion for infinitely many factors, each of which is an infinite join.\nThe result is a join over all ways of choosing one term from each factor, just as $(a_1 + a_2)(b_1 + b_2)$ is the sum of the four products $a_i b_j$.\nIn the lattice of subsets of a set, with unions and intersections, this expansion always works.\n",
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