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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nFor a function $\\rho \\colon X \\to \\R$, its support $\\operatorname{supp} \\rho$ is the closure of $\\set{x \\in X : \\rho(x) \\neq 0}$.\nA *partition of unity* on $X$ is a family $(\\rho_i)_{i \\in I}$ of [[continuous functions|0X3WGX95FE]] $\\rho_i \\colon X \\to [0, 1]$ such that:\n\n1. the family of supports $(\\operatorname{supp} \\rho_i)_{i \\in I}$ is [[locally finite|DZ297MMGV8]]: each point of $X$ has a neighbourhood that meets only finitely many of them;\n2. $\\sum_{i \\in I} \\rho_i(x) = 1$ for every $x \\in X$.\n\nLet $(U_i)_{i \\in I}$ be an [[open cover|049PXYHW1T]] of $X$ with the same index set.\nThe partition of unity is *subordinate* to $(U_i)_{i \\in I}$ if $\\operatorname{supp} \\rho_i \\subset U_i$ for every $i$.\nIf $X$ is a [[smooth manifold|F45VQ47GPV]] and every $\\rho_i$ is smooth, it is a *smooth partition of unity*.\n\n## Remarks\n\nExample on $\\R$ with the cover $U_1 = (-\\infty, 1)$ and $U_2 = (-1, \\infty)$.\nLet $\\rho_1(x) = 1$ for $x \\le -1/2$, $\\rho_1(x) = 1/2 - x$ for $-1/2 \\le x \\le 1/2$, and $\\rho_1(x) = 0$ for $x \\ge 1/2$, and let $\\rho_2 = 1 - \\rho_1$.\nBoth are continuous with values in $[0, 1]$, the sum is $1$, and $\\operatorname{supp} \\rho_1 = (-\\infty, 1/2] \\subset U_1$ and $\\operatorname{supp} \\rho_2 = [-1/2, \\infty) \\subset U_2$.\nReplacing the linear piece by a smooth monotone transition gives a smooth partition of unity.\n\nNon-example: the constant functions $\\rho_1 = \\rho_2 = 1/4$ on $\\R$ are continuous with locally finite supports, but their sum is $1/2$, so clause 2 fails.\n\nClause 1 makes the sum in clause 2 a finite sum near each point, so it is continuous, or smooth in the smooth case.\nEvery paracompact Hausdorff space has a partition of unity subordinate to any open cover.\nEvery smooth manifold has a smooth partition of unity subordinate to any open cover; the proof uses bump functions and second countability.\n\n## Explanations\n\n### L1 human\n\nA partition of unity is a family of continuous functions with values in $[0,1]$ that add up to $1$, where near each point only finitely many of them are nonzero.\n",
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