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  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]] and let $X$ be a [[topological space|0CQHFPGJ45]].\nA *topological transformation group* is a triple $(G, X, \\alpha)$ where $\\alpha \\colon G \\times X \\to X$ is a [[group action|0WCY42EV9V]] of $G$ on $X$, written $\\alpha(g, x) = g \\cdot x$, such that $\\alpha$ is [[continuous|0VGFHTM75Y]].\nHere $G \\times X$ carries the [[product topology|0GBG8B49AA]].\nOne also says that $G$ acts continuously on $X$.\n\n## Remarks\n\nExample: the group $\\mathrm{SO}(3)$ of rotations of $\\R^3$, with the topology from the matrix entries, acting on the unit sphere $S^2$ by $(A, x) \\mapsto Ax$.\nThe map is given by polynomials in the entries of $A$ and the coordinates of $x$, so it is continuous.\nNon-example: $(\\R, +)$ with the usual topology acting on $\\R$ with the discrete topology by $(t, x) \\mapsto t + x$.\nThis is a group action, but it is not continuous: for fixed $x$, the map $t \\mapsto t + x$ from $\\R$ to the discrete space is not continuous.\nFor each $g \\in G$ the map $x \\mapsto g \\cdot x$ is a homeomorphism of $X$, with inverse $x \\mapsto g^{-1} \\cdot x$.\n",
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