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  "body": "\n## Definition\n\nLet $(X, x_0)$ be a [[pointed space|T9DG8DDE75]] and let $I = [0, 1]$.\nThe *path space* of $(X, x_0)$ is\n$$PX = \\set{\\gamma \\colon I \\to X \\text{ continuous} : \\gamma(0) = x_0},$$\nthe set of [[paths|03NAMV1DHJ]] starting at $x_0$, with the [[subspace topology|0WS9P12FQR]] of the compact-open topology.\nThe *path space fibration* is the map\n$$p \\colon PX \\to X, \\quad p(\\gamma) = \\gamma(1).$$\n\n## Remarks\n\nThe map $p$ is a Hurewicz fibration, and its fiber $p^{-1}(x_0)$ is the [[loop space|2JFF5F3EGF]] $\\Omega X$.\nThe space $PX$ is contractible: $h_s(\\gamma)(t) = \\gamma((1 - s) t)$ deforms every path to the constant path at $x_0$.\nSo the long exact sequence of homotopy groups gives $\\pi_n(\\Omega X) \\cong \\pi_{n+1}(X, x_0)$ for $n \\ge 0$.\n\nExample: for $X = \\R$ and $x_0 = 0$, the fiber over $a \\in \\R$ is the space of paths from $0$ to $a$, and $p$ is surjective, since the path $t \\mapsto t a$ ends at $a$.\n\nNon-example: if $X$ is not path-connected, $p$ is not surjective.\nFor $X = \\set{0, 1}$ with the discrete topology and $x_0 = 0$, every path from $0$ is constant, so $PX$ is a point and $p^{-1}(1)$ is empty.\n\nThe fiber of $p$ over $y \\in X$ is the [[homotopy fiber|4F3KWXYXBA]] over $y$ of the inclusion $\\set{x_0} \\to X$.\nSo $p$ replaces this inclusion by a fibration with a contractible total space.\n",
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