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  "body": "\n## Summary\n\nAn Esakia space is a [[Priestley space|ZGBV8V3823]] in which the down-set of every clopen set is again clopen.\nEsakia spaces are the spaces dual to [[Heyting algebras|NG943XNQWH]], by [[Esakia duality|CD9Y7YP4EJ]].\nThe Heyting algebra of an Esakia space is its algebra of clopen up-sets.\nThey connect the algebraic semantics of [[intuitionistic logic|Q11N8X99FK]] with its [[Kripke semantics|NGHWZ9Y7ZV]].\n\n## Explanations\n\n### L1 human\n\nAn Esakia space is a compact space whose points are also partially ordered, in a way that fits the topology well.\nOne can think of the points as states of knowledge, where moving up in the order means learning more.\nThe extra rule says that the set of states below a clopen set is again clopen.\nThis rule makes the implication of two clopen up-sets again a clopen up-set.\nEvery finite partially ordered set, with the discrete topology, is an Esakia space.\n",
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