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  "body": "\n## Definition\n\nFor $n \\ge 0$ let $[n] = \\set{0, 1, \\dots, n}$, and let\n$$\\abs{\\Delta^n} = \\set{(t_0, \\dots, t_n) \\in \\R^{n+1} : t_i \\ge 0, \\ t_0 + \\cdots + t_n = 1}$$\nbe the standard topological $n$-simplex, with vertices $e_0, \\dots, e_n$.\nAn order-preserving map $\\theta \\colon [m] \\to [n]$ induces the affine map $\\theta_* \\colon \\abs{\\Delta^m} \\to \\abs{\\Delta^n}$ with $\\theta_*(e_i) = e_{\\theta(i)}$.\nLet $X$ be a [[topological space|0CNQXVBHCX]].\nThe *singular simplicial set* $\\operatorname{Sing} X$ is the simplicial set with:\n\n1. $(\\operatorname{Sing} X)_n$ the set of [[continuous maps|0SY9ECQ19Q]] $\\sigma \\colon \\abs{\\Delta^n} \\to X$, called singular $n$-simplices.\n2. For $\\theta \\colon [m] \\to [n]$ order-preserving, the map $\\theta^* \\colon (\\operatorname{Sing} X)_n \\to (\\operatorname{Sing} X)_m$ given by $\\theta^*(\\sigma) = \\sigma \\circ \\theta_*$.\n\n## Remarks\n\nThe identities $(\\theta \\circ \\psi)_* = \\theta_* \\circ \\psi_*$ and $\\id_* = \\id$ show that $\\operatorname{Sing} X$ is a functor $\\Delta^{\\mathrm{op}} \\to \\mathrm{Set}$, that is, a simplicial set.\nA continuous map $f \\colon X \\to Y$ induces $\\operatorname{Sing} f$ by $\\sigma \\mapsto f \\circ \\sigma$, so $\\operatorname{Sing}$ is a functor from spaces to simplicial sets.\n\nExample: if $X$ is a single point, each $(\\operatorname{Sing} X)_n$ has exactly one element, so $\\operatorname{Sing} X$ is the terminal simplicial set $\\Delta^0$.\n\nNon-example: the simplicial set whose $n$-simplices are all functions $\\abs{\\Delta^n} \\to X$, continuous or not, is a simplicial set, but it is not $\\operatorname{Sing} X$ for $X = [0, 1]$.\nIt contains discontinuous functions $\\abs{\\Delta^1} \\to [0, 1]$, which clause 1 excludes.\n\n$\\operatorname{Sing} X$ is always a Kan complex.\n",
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