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  "body": "\n## Definition\n\nLet $X$ and $Y$ be topological spaces and let $f, g \\colon X \\to Y$ be [[continuous maps|0VGFHTM75Y]].\nThe maps $f$ and $g$ are *homotopic*, written $f \\simeq g$, if there is a [[homotopy|9YA6S6HD9V]] from $f$ to $g$.\n\n## Remarks\n\nThe relation $\\simeq$ is an [[equivalence relation|0G8VEZ3WTK]] on the set of continuous maps $X \\to Y$.\nReflexivity uses the constant homotopy $H(x, t) = f(x)$, symmetry reverses time, and transitivity runs two homotopies one after the other, each at double speed.\nThe equivalence class of $f$ is its *homotopy class* $[f]$, and the set of classes is written $[X, Y]$.\nHomotopy is compatible with composition: if $f \\simeq f'$ and $g \\simeq g'$, then $g \\circ f \\simeq g' \\circ f'$.\nExample: the maps $f(x) = x$ and $g(x) = 0$ from $[0, 1]$ to $\\R$ are homotopic via $H(x, t) = (1 - t)x$.\nNon-example: the identity of the [[circle|060W5CBZP6]] $S^1$ and a constant map $S^1 \\to S^1$ are not homotopic, because they induce different maps on the [[fundamental group|S1XK3HXHFX]].\n",
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