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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]].\nAn *interior operator* on $P$ is a map $i \\colon P \\to P$ such that, for all $x, y \\in P$:\n\n1. $i(x) \\le x$ (deflationary);\n2. if $x \\le y$, then $i(x) \\le i(y)$ (monotone);\n3. $i(i(x)) = i(x)$ (idempotent).\n\nAn element $x$ with $i(x) = x$ is *open*.\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 [[interior|0C38S0KMJK]] is an interior operator: the interior lies in $A$, grows with $A$, and is open, so taking it twice changes nothing.\nOn $\\R$, the map $i(x) = x - 1$ is deflationary and monotone but not idempotent, since $i(i(0)) = -2 \\ne -1 = i(0)$.\nA map $i$ is an interior operator on $P$ if and only if it is a [[closure operator|17FVAN6HXX]] on the [[dual|BS1RD4F5MD]] of $P$.\nOn a power set, $A \\mapsto X \\setminus c(X \\setminus A)$ is an interior operator for each closure operator $c$.\nOther names are kernel operator and dual closure operator.\n\n## Explanations\n\n### L1 human\n\nAn interior operator moves each element down to the nearest open element.\nOpen elements stay where they are.\n",
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