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  "body": "\n## Definition\n\nLet $(X, x_0)$ be a [[pointed space|T9DG8DDE75]] and let $I = [0, 1]$.\nA *loop* at $x_0$ is a continuous map $\\gamma \\colon I \\to X$ with $\\gamma(0) = \\gamma(1) = x_0$.\nFor loops $\\alpha$ and $\\beta$ at $x_0$, the *concatenation* $\\alpha \\cdot \\beta$ is the loop with $(\\alpha \\cdot \\beta)(s) = \\alpha(2s)$ for $0 \\le s \\le 1/2$ and $(\\alpha \\cdot \\beta)(s) = \\beta(2s - 1)$ for $1/2 \\le s \\le 1$.\nThe *fundamental group* $\\pi_1(X, x_0)$ is the set of [[path homotopy|CCNKA8C711]] classes $[\\gamma]$ of loops at $x_0$, with the product $[\\alpha][\\beta] = [\\alpha \\cdot \\beta]$.\n\n## Remarks\n\nThe product is well defined, and it makes $\\pi_1(X, x_0)$ a [[group|G5S1M9]].\nThe identity is the class of the constant loop, and the inverse of $[\\gamma]$ is the class of the reversed loop $s \\mapsto \\gamma(1 - s)$.\nA pointed map $f \\colon (X, x_0) \\to (Y, y_0)$ induces a homomorphism $f_* \\colon \\pi_1(X, x_0) \\to \\pi_1(Y, y_0)$, $[\\gamma] \\mapsto [f \\circ \\gamma]$.\nIf $X$ is [[path-connected|0WB6DH499T]], the group does not depend on $x_0$ up to isomorphism.\nExample: for a convex set $C \\subseteq \\R^n$ and $x_0 \\in C$, every loop is path homotopic to the constant loop by the straight-line homotopy, so $\\pi_1(C, x_0)$ is trivial.\nNon-example: loops themselves, without passing to classes, do not form a group under concatenation.\nIndeed $(\\alpha \\cdot \\beta) \\cdot \\gamma$ and $\\alpha \\cdot (\\beta \\cdot \\gamma)$ are in general different maps, which agree only up to path homotopy.\nA path-connected space with trivial fundamental group is *simply connected*.\n",
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