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  "body": "\n## Summary\n\nA closure operator on a [[partially ordered set|011C29Q1S1]] is a map $c$ that is extensive, monotone and idempotent.\nIt sends each element to the least \"closed\" element above it.\nExamples on power sets are topological [[closure|0DRKT1QJR7]], the [[linear span|0HR573CHNM]] of vectors and the [[subgroup generated by|0RG9SPQSWM]] a set.\nClosure operators on a power set correspond to [[Moore families|9ZVCJWYPB2]].\n\n## Explanations\n\n### L1 human\n\nA closure operator completes an object to the smallest object of a special kind that contains it.\nIn a vector space, take a set of vectors and replace it by all their linear combinations.\nDoing it again adds nothing new, a bigger set gives a bigger span, and the span always contains the original set.\nThese three features, growing, keeping order, and stopping after one step, define a closure operator.\n",
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