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  "body": "\n## Definition\n\nLet $X$ and $Y$ be topological spaces.\nA [[continuous map|0VGFHTM75Y]] $f \\colon X \\to Y$ is a *homotopy equivalence* if there is a continuous map $g \\colon Y \\to X$ with these two properties.\nThe map $g \\circ f$ is [[homotopic|F2F143KQ2M]] to $\\id_X$, and $f \\circ g$ is homotopic to $\\id_Y$.\nSuch a map $g$ is a *homotopy inverse* of $f$.\n\n## Remarks\n\nEvery homeomorphism is a homotopy equivalence, with its inverse as homotopy inverse.\nExample: let $i \\colon S^1 \\to \\R^2 \\setminus \\set{0}$ be the inclusion and $r(x) = x / \\norm{x}$.\nThen $r \\circ i = \\id_{S^1}$, and $H(x, t) = (1 - t) x + t x / \\norm{x}$ is a homotopy in $\\R^2 \\setminus \\set{0}$ from $\\id$ to $i \\circ r$, so $i$ is a homotopy equivalence.\nNon-example: the constant map $S^1 \\to \\set{p}$ is not a homotopy equivalence, since the [[fundamental group|S1XK3HXHFX]] of $S^1$ is $\\Z$ and that of a point is trivial.\nComposites of homotopy equivalences are homotopy equivalences.\n",
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