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  "body": "\n## Definition\n\nLet $(X, x_0)$ be a [[pointed space|T9DG8DDE75]] and let $I = [0, 1]$.\nThe *loop space* of $(X, x_0)$ is the set\n$$\\Omega X = \\set{\\gamma \\colon I \\to X \\text{ continuous} : \\gamma(0) = \\gamma(1) = x_0},$$\nwith the [[subspace topology|0WS9P12FQR]] of the [[compact-open topology|RG15F0TWPW]] on the set of continuous maps $I \\to X$.\nIts base point is the constant loop at $x_0$.\n\n## Remarks\n\nConcatenation $(\\alpha \\cdot \\beta)(s) = \\alpha(2s)$ for $s \\le 1/2$ and $\\beta(2s - 1)$ for $s \\ge 1/2$ is a continuous product on $\\Omega X$ that is associative and unital up to homotopy.\nThe set of path components $\\pi_0(\\Omega X)$ is in bijection with $\\pi_1(X, x_0)$, and $\\pi_n(\\Omega X) \\cong \\pi_{n+1}(X, x_0)$ for $n \\ge 1$.\nFor pointed spaces $A$ and $X$, pointed homotopy classes of maps $\\Sigma A \\to X$ from the reduced [[suspension|Q7RFNJH76Q]] correspond bijectively to pointed homotopy classes of maps $A \\to \\Omega X$.\nExample: if $X$ is the one-point space, then $\\Omega X$ is a single point, the constant loop.\nNon-example: the free loop space of all continuous maps $S^1 \\to X$ with no condition at a base point is a different space; it contains loops that miss $x_0$.\n",
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