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  "body": "\n## Definition\n\nA *topological monoid* is a [[monoid|0N6R902FG1]] $M$ together with a [[topology|0CQHFPGJ45]] on $M$ such that the multiplication $M \\times M \\to M$, $(x, y) \\mapsto xy$, is a [[continuous multiplication|ZZK2GM1V15]].\nThat is, the map is continuous when $M \\times M$ carries the product topology.\n\n## Remarks\n\nExample: $(\\R, \\cdot)$ with the usual topology.\nMultiplication is associative with identity $1$, and $(x, y) \\mapsto xy$ is continuous on $\\R^2$.\nNon-example: the monoid $(\\Z, +)$ with the cofinite topology.\nIt has identity $0$, but addition is not jointly continuous, because every nonempty open set $U \\times V$ of $\\Z \\times \\Z$ contains a pair $(x, -x)$, so the preimage of $\\Z \\setminus \\set{0}$ is not open.\nNo condition on the identity element is needed beyond its existence.\nA topological monoid in which every element is invertible and inversion is continuous is a topological group.\n",
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