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  "body": "\n## Definition\n\nA *paratopological group* is a [[group|G5S1M9]] $G$ together with a [[topology|0CQHFPGJ45]] on $G$ such that the multiplication $G \\times G \\to G$, $(x, y) \\mapsto xy$, is a [[continuous multiplication|ZZK2GM1V15]].\nNo condition is placed on the inversion $x \\mapsto x^{-1}$.\n\n## Remarks\n\nExample: $(\\R, +)$ with the topology of the [[Sorgenfrey line|Y4T3GVK8GV]], generated by the intervals $[a, b)$.\nIf $x + y \\in [c, d)$, choose $\\varepsilon \u003e 0$ with $x + y + 2\\varepsilon \u003c d$; then $[x, x + \\varepsilon) + [y, y + \\varepsilon) \\subseteq [x + y, x + y + 2\\varepsilon) \\subseteq [c, d)$, so addition is continuous.\nInversion is not continuous, since the preimage of $[0, 1)$ under $x \\mapsto -x$ is $(-1, 0]$, which is not open.\nSo this is a paratopological group that is not a [[topological group|92700QAB5Y]].\nNon-example: $(\\Z, +)$ with the cofinite topology is a group, but addition is not jointly continuous, because every nonempty open set $U \\times V$ of $\\Z \\times \\Z$ contains a pair $(x, -x)$.\nA paratopological group is the same as a topological monoid whose underlying monoid is a group.\n",
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