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  "body": "\n## Definition\n\nA *topological group* is a [[group|G5S1M9]] $G$ together with a [[topology|0CQHFPGJ45]] on $G$ such that:\n\n1. the multiplication $G \\times G \\to G$, $(x, y) \\mapsto xy$, is a [[continuous multiplication|ZZK2GM1V15]], where $G \\times G$ carries the product topology;\n2. the inversion $G \\to G$, $x \\mapsto x^{-1}$, is [[continuous|0VGFHTM75Y]].\n\n## Remarks\n\nExample: $(\\R, +)$ with the usual topology.\nAddition $(x, y) \\mapsto x + y$ is continuous on $\\R^2$, and $x \\mapsto -x$ is continuous.\nNon-example: $(\\R, +)$ with the lower limit topology, generated by the intervals $[a, b)$.\nAddition is still continuous, but inversion fails clause 2: the preimage of the open set $[0, 1)$ under $x \\mapsto -x$ is $(-1, 0]$, which is not open.\nThe two clauses together are equivalent to the single condition that $(x, y) \\mapsto x y^{-1}$ is continuous from $G \\times G$ to $G$.\nEvery group with the discrete topology, and every group with the indiscrete topology, is a topological group.\nMany authors also require the topology to be Hausdorff.\n",
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