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  "body": "\n## Summary\n\nA Heyting algebra is a bounded lattice with an implication operation $a \\to b$, the largest element whose meet with $a$ lies below $b$.\nEvery Heyting algebra is a [[distributive lattice|0HM0HQCK6N]], and every [[Boolean algebra|0XADJKM699]] is a Heyting algebra.\nHeyting algebras are the algebraic models of [[intuitionistic logic|Q11N8X99FK]], in the same way that Boolean algebras are the models of classical logic.\nThe open sets of any topological space form a Heyting algebra.\n\n## Explanations\n\n### L1 human\n\nIn a Heyting algebra one can form \"and\", \"or\" and \"implies\", but \"not $a$\" need not satisfy \"$a$ or not $a$\".\nThe main picture is the family of open sets of the plane or the line.\n\"And\" is intersection, \"or\" is union, and \"not $U$\" is the interior of the complement of $U$.\nOn the real line, the open set $U = \\R \\setminus \\set{0}$ has \"not $U$\" equal to the empty set, so \"$U$ or not $U$\" is $\\R \\setminus \\set{0}$, not the whole line.\nThis failure of the law of excluded middle is exactly what intuitionistic logic allows.\n",
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