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  "body": "\n## Summary\n\nA set with a binary operation and a topology has continuous multiplication when the operation, seen as a map of two variables, is continuous.\nHere the domain carries the [[product topology|0GBG8B49AA]].\nThis single condition is the common base of [[topological semigroups|QXB6JDG68F]], [[topological monoids|RA56GHB9JZ]] and [[topological groups|92700QAB5Y]].\nIt asks for joint continuity, which is stronger than continuity in each variable separately.\n\n## Explanations\n\n### L1 human\n\nThink of multiplying two numbers $x$ and $y$.\nIf you move $x$ and $y$ a little, the product $xy$ also moves only a little.\nThat is what continuous multiplication means: small changes in both inputs give a small change in the output.\nFor example, ordinary multiplication on the real line has this property, so $(\\R, \\cdot)$ has continuous multiplication.\n",
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