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  "body": "\n## Definition\n\nLet $X$ be a topological space and let $I = [0, 1]$ be the unit [[interval|0Q1D85B8G6]].\nA *path* in $X$ is a continuous map $\\gamma \\colon I \\to X$, from $\\gamma(0)$ to $\\gamma(1)$.\nLet $\\gamma_0, \\gamma_1 \\colon I \\to X$ be paths with $\\gamma_0(0) = \\gamma_1(0) = x_0$ and $\\gamma_0(1) = \\gamma_1(1) = x_1$.\nA *path homotopy* from $\\gamma_0$ to $\\gamma_1$ is a continuous map $H \\colon I \\times I \\to X$ such that:\n\n1. $H(s, 0) = \\gamma_0(s)$ and $H(s, 1) = \\gamma_1(s)$ for all $s \\in I$;\n2. $H(0, t) = x_0$ and $H(1, t) = x_1$ for all $t \\in I$.\n\nThe paths are *path homotopic*, written $\\gamma_0 \\simeq_p \\gamma_1$, if a path homotopy exists.\n\n## Remarks\n\nA path homotopy is the same as a [[homotopy relative|ADDS6HE5SV]] to $\\set{0, 1}$.\nPath homotopy is an equivalence relation on paths from $x_0$ to $x_1$.\nExample: in a convex set $C \\subseteq \\R^n$, the map $H(s, t) = (1 - t)\\gamma_0(s) + t \\gamma_1(s)$ is a path homotopy, since $H(0, t) = x_0$ and $H(1, t) = x_1$.\nNon-example: in $S^1$, the loop $s \\mapsto e^{2\\pi i s}$ and the constant loop at $1$ are not path homotopic; the [[fundamental group of the circle|6SQXPE0JPN]] tells them apart.\n",
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