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  "body": "\n## Definition\n\nA *topological semigroup* is a [[semigroup|0XMEM406JM]] $S$ together with a [[topology|0CQHFPGJ45]] on $S$ such that the multiplication $S \\times S \\to S$, $(x, y) \\mapsto xy$, is a [[continuous multiplication|ZZK2GM1V15]].\nThat is, the map is continuous when $S \\times S$ carries the product topology.\n\n## Remarks\n\nExample: $[0, 1]$ with ordinary multiplication and the topology from $\\R$.\nThe product is associative, it maps $[0,1] \\times [0,1]$ into $[0,1]$, and it is continuous as the restriction of a polynomial.\nNon-example: the semigroup $(\\Z, +)$ with the cofinite topology.\nAddition is associative, but it is not jointly continuous: the preimage of $\\Z \\setminus \\set{0}$ is not open, since every nonempty basic open set $U \\times V$ contains a pair $(x, -x)$.\nIf one asks only that each translation $y \\mapsto xy$ and $y \\mapsto yx$ be continuous, the weaker structure is called a semitopological semigroup.\n",
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