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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], let $I = [0, 1]$, and let $f, g \\colon X \\to Y$ be [[continuous maps|0VGFHTM75Y]].\nA *homotopy* from $f$ to $g$ is a continuous map $H \\colon X \\times I \\to Y$, where $X \\times I$ carries the [[product topology|0GBG8B49AA]], such that $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$ for all $x \\in X$.\n\n## Remarks\n\nFor each $t \\in I$, the map $h_t = H(\\cdot, t) \\colon X \\to Y$ is continuous, so a homotopy is a continuous family $(h_t)_{t \\in I}$ of maps with $h_0 = f$ and $h_1 = g$.\nExample: for $X = Y = \\R$, $f = \\id$ and $g = 0$, the map $H(x, t) = (1 - t)x$ is continuous, with $H(x, 0) = x$ and $H(x, 1) = 0$, so it is a homotopy from $f$ to $g$.\nNon-example: for the same $f$ and $g$, the map $K(x, t) = x$ for $t \u003c 1$ and $K(x, 1) = 0$ has the right ends but is not continuous at $(1, 1)$, so it is not a homotopy.\nIf $H$ is a homotopy from $f$ to $g$, then $(x, t) \\mapsto H(x, 1 - t)$ is a homotopy from $g$ to $f$.\n",
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