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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]].\nFor $n \\ge 0$ let $\\Delta^n = \\set{(t_0, \\dots, t_n) \\in \\R^{n+1} : t_i \\ge 0, \\ \\sum_i t_i = 1}$ be the standard $n$-simplex, and for $0 \\le i \\le n$ let $\\delta^i \\colon \\Delta^{n-1} \\to \\Delta^n$ be the map that inserts a $0$ in position $i$.\nA *singular $n$-simplex* in $X$ is a [[continuous map|0SY9ECQ19Q]] $\\sigma \\colon \\Delta^n \\to X$.\nLet $S_n(X)$ be the [[free abelian group|0WJX8MGG37]] on the singular $n$-simplices, and $S_n(X) = 0$ for $n \u003c 0$.\nThe boundary map $\\partial_n \\colon S_n(X) \\to S_{n-1}(X)$ is given on generators by\n$$\\partial_n \\sigma = \\sum_{i=0}^{n} (-1)^i \\, \\sigma \\circ \\delta^i,$$\nand $\\partial_0 = 0$.\nThen $\\partial_n \\circ \\partial_{n+1} = 0$, so $S_\\bullet(X)$ is a chain complex.\nThe *$n$-th singular homology group* of $X$ is\n$$H_n(X) = \\ker \\partial_n / \\operatorname{im} \\partial_{n+1}.$$\n\n## Remarks\n\nA continuous map $f \\colon X \\to Y$ induces $f_* \\colon H_n(X) \\to H_n(Y)$ by $\\sigma \\mapsto f \\circ \\sigma$, and homotopic maps induce the same homomorphism.\nHomology with coefficients in an abelian group $A$ uses $S_n(X) \\otimes A$.\n\nExample: let $X$ be a point.\nThere is exactly one singular $n$-simplex $\\sigma_n$ for each $n$, and $\\partial_n \\sigma_n = \\sum_{i=0}^n (-1)^i \\sigma_{n-1}$, which is $\\sigma_{n-1}$ for even $n \\ge 2$ and $0$ for odd $n$.\nSo $\\partial_n$ is an isomorphism for even $n \\ge 2$ and zero for odd $n$, which gives $H_0(X) \\cong \\Z$ and $H_n(X) = 0$ for $n \\ge 1$.\n\nNon-example: using all functions $\\Delta^n \\to X$, continuous or not, in place of singular simplices still gives a chain complex, but not singular homology.\nAny two points of $X$ are then joined by a $1$-simplex, so its $H_0$ is $\\Z$ for every nonempty $X$.\nFor $X$ a discrete space with two points, singular homology gives $H_0(X) \\cong \\Z^2$.\n",
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