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  "body": "\n## Definition\n\nFor $n \\ge 0$ let $\\Delta^n$ be the standard $n$-simplex with ordered vertices $e_0, \\dots, e_n$, and let $\\mathring{\\Delta}^n$ be its interior.\nFor $0 \\le i \\le n$ with $n \\ge 1$, let $\\delta^i \\colon \\Delta^{n-1} \\to \\Delta^n$ be the affine map onto the $i$-th face that sends $e_0, \\dots, e_{n-1}$ in order to the vertices $e_0, \\dots, \\widehat{e_i}, \\dots, e_n$.\nLet $X$ be a [[topological space|0CNQXVBHCX]].\nA *$\\Delta$-complex structure* on $X$ is a family of [[continuous maps|0SY9ECQ19Q]] $\\sigma_\\alpha \\colon \\Delta^{n_\\alpha} \\to X$ such that:\n\n1. Each $\\sigma_\\alpha$ is injective on $\\mathring{\\Delta}^{n_\\alpha}$, and each point of $X$ lies in $\\sigma_\\alpha(\\mathring{\\Delta}^{n_\\alpha})$ for exactly one $\\alpha$.\n2. For each $\\alpha$ with $n_\\alpha \\ge 1$ and each $i$, the map $\\sigma_\\alpha \\circ \\delta^i$ is $\\sigma_\\beta$ for some $\\beta$ with $n_\\beta = n_\\alpha - 1$.\n3. A subset $A \\subseteq X$ is open if and only if $\\sigma_\\alpha^{-1}(A)$ is open in $\\Delta^{n_\\alpha}$ for every $\\alpha$.\n\nA *$\\Delta$-complex* is a space with a $\\Delta$-complex structure.\n\n## Remarks\n\nEvery $\\Delta$-complex is a CW complex whose cells are the open simplices $\\sigma_\\alpha(\\mathring{\\Delta}^{n_\\alpha})$.\nThe combinatorial data of a $\\Delta$-complex, the sets of $n$-simplices with the face maps $d_i$ given by clause 2, form a semi-simplicial set.\n\nExample: the circle $S^1$ with one $0$-simplex $\\sigma_0 \\colon \\Delta^0 \\to S^1$ onto a point $p$ and one $1$-simplex $\\sigma_1 \\colon \\Delta^1 \\to S^1$ that wraps once around, sending both ends to $p$.\nClause 1 holds since $\\sigma_1$ is injective on the open interval and misses only $p$.\nClause 2 holds since both faces of $\\sigma_1$ equal $\\sigma_0$, and clause 3 holds since $S^1$ is a quotient of $\\Delta^1$.\n\nNon-example: the circle with one $0$-simplex at $p$ and one $1$-simplex that wraps twice around fails clause 1, since the map is not injective on the open interval.\n",
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