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  "body": "\n## Definition\n\nA *semitopological group* is a [[group|G5S1M9]] $G$ together with a [[topology|0CQHFPGJ45]] on $G$ such that the multiplication $G \\times G \\to G$ is separately continuous.\nThat is, for each $a \\in G$ the left translation $x \\mapsto ax$ and the right translation $x \\mapsto xa$ are [[continuous|0VGFHTM75Y]] maps $G \\to G$.\n\n## Remarks\n\nExample: $(\\Z, +)$ with the cofinite topology.\nEach translation $x \\mapsto x + a$ is a bijection, so it maps cofinite sets to cofinite sets and is continuous.\nThis group is not a [[topological group|92700QAB5Y]]: every nonempty basic open set $U \\times V$ of $\\Z \\times \\Z$ contains a pair $(x, -x)$, so the preimage of $\\Z \\setminus \\set{0}$ under addition is not open.\nNon-example: the multiplicative group $\\R \\setminus \\set{0}$ with the topology whose open sets are $\\emptyset$, the whole set, and the sets $(a, \\infty) \\setminus \\set{0}$ for $a \\in \\R$.\nThe left translation $x \\mapsto (-1) x$ is not continuous: the preimage of the open set $(1, \\infty)$ is $(-\\infty, -1)$, which is not open.\nEvery topological group is a semitopological group, since joint continuity of multiplication implies separate continuity.\nIn a semitopological group each translation is a homeomorphism, with inverse the translation by $a^{-1}$.\n",
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