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  "body": "\n## Definition\n\nLet $X$ and $Y$ be topological spaces, and let $f \\colon X \\to Y$ be a [[continuous map|0VGFHTM75Y]].\nThe map $f$ is a *weak homotopy equivalence* if both of the following hold:\n\n1. $f$ induces a bijection $\\pi_0(X) \\to \\pi_0(Y)$ between the sets of path components;\n2. for every $x \\in X$ and every $n \\ge 1$, the induced map $f_* \\colon \\pi_n(X, x) \\to \\pi_n(Y, f(x))$ of [[homotopy groups|P976WYEFMB]] is an isomorphism.\n\n## Remarks\n\nExample: every homotopy equivalence $f$ with homotopy inverse $g$ is a weak homotopy equivalence, since $g_*$ inverts $f_*$ up to the change-of-base-point isomorphisms along the homotopies.\n\nExample: all homotopy groups of the [[Warsaw circle|17VG475BRV]] are trivial, so the map from it to a point is a weak homotopy equivalence.\nThis map is not a homotopy equivalence.\n\nNon-example: the inclusion $S^1 \\hookrightarrow D^2$ of the boundary circle into the disk is a bijection on $\\pi_0$.\nIt fails clause 2, because $\\pi_1(S^1) \\cong \\Z$ while $\\pi_1(D^2)$ is trivial.\n\nWeak homotopy equivalences induce isomorphisms on singular homology and cohomology.\nA weak homotopy equivalence is also called a weak equivalence.\n",
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