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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]].\nA *closure operator* on $P$ is a map $c \\colon P \\to P$ such that, for all $x, y \\in P$:\n\n1. $x \\le c(x)$ (extensive);\n2. if $x \\le y$, then $c(x) \\le c(y)$ (monotone);\n3. $c(c(x)) = c(x)$ (idempotent).\n\nAn element $x$ with $c(x) = x$ is *closed*.\n\n## Remarks\n\nLet $X$ be a topological space and $P$ the set of subsets of $X$ ordered by inclusion.\nThe map sending $A$ to its [[closure|0DRKT1QJR7]] $\\overline{A}$ is a closure operator: $A \\subseteq \\overline{A}$, $A \\subseteq B$ gives $\\overline{A} \\subseteq \\overline{B}$, and $\\overline{\\overline{A}} = \\overline{A}$.\nOn $\\R$, the map $c(x) = x + 1$ is extensive and monotone but not idempotent, since $c(c(0)) = 2 \\ne 1 = c(0)$.\nThe three clauses together are equivalent to one condition: $x \\le c(y)$ if and only if $c(x) \\le c(y)$, for all $x, y \\in P$.\nThe closed elements determine $c$: $c(x)$ is the least closed element above $x$.\nIf $(f, g)$ is a [[Galois connection|MBVJ4JQXC5]] with lower adjoint $f$, then $g \\circ f$ is a closure operator.\nThe dual notion is an [[interior operator|BPXY5A22BB]].\n\n## Explanations\n\n### L1 human\n\nA closure operator moves each element up to the nearest closed element.\nClosed elements stay where they are.\n",
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