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  "body": "\n## Summary\n\nA topological group is a [[group|G5S1M9]] with a [[topology|0CQHFPGJ45]] in which multiplication and inversion are continuous.\nIt is the basic object where algebra and topology work together.\nIt is the setting of abstract harmonic analysis and of the theory of Lie groups.\nTranslations by group elements are homeomorphisms, so the topology near any point looks like the topology near the identity.\nThe real line under addition, the circle group, matrix groups and the $p$-adic integers are all topological groups.\n\n## Explanations\n\n### L1 human\n\nA topological group is a group where it also makes sense to say that two elements are close.\nThe group law must respect closeness: if $x$ is near $x'$ and $y$ is near $y'$, then $xy$ is near $x'y'$, and $x^{-1}$ is near $x'^{-1}$.\nPicture the real line under addition: shifting the whole line by a number does not tear it.\nAnother example is the circle of complex numbers of absolute value $1$ under multiplication, where multiplying means rotating.\n",
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