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  "body": "\n## Definition\n\nTwo topological spaces $X$ and $Y$ are *homotopy equivalent*, or *have the same homotopy type*, written $X \\simeq Y$, if there is a [[homotopy equivalence|XZG6MTSWCN]] $f \\colon X \\to Y$.\nThe *homotopy type* of $X$ is the class of all spaces homotopy equivalent to $X$.\n\n## Remarks\n\nThe relation $\\simeq$ is reflexive, symmetric and transitive, so it behaves as an [[equivalence relation|0G8VEZ3WTK]] on topological spaces.\nSymmetry holds because a homotopy inverse is itself a homotopy equivalence.\nExample: $\\R^n$ has the homotopy type of a point; the maps $\\R^n \\to \\set{0}$ and $\\set{0} \\to \\R^n$ are mutually inverse up to the homotopy $H(x, t) = (1 - t) x$.\nNon-example: $S^1$ and a point do not have the same homotopy type, since their [[fundamental groups|S1XK3HXHFX]] differ.\nHomeomorphic spaces have the same homotopy type, but the converse fails, as $\\R$ and a point show.\n",
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