[proofplan] We first use the paracompactness and regularity properties of smooth manifolds to replace the given cover by a locally finite family of relatively compact coordinate balls whose closures lie inside members of the original cover. On each such coordinate ball we construct a smooth bump function, positive on a smaller ball and supported in the larger one. The locally finite sum of these bump functions is smooth and everywhere positive, so division by this sum produces a partition of unity indexed by the refining balls. Finally we group the refining functions according to the original [open set](/page/Open%20Set) containing their supports. [/proofplan] [step:Choose a locally finite coordinate refinement whose closures lie in the original cover] Let $n := \dim M$. For $q \in \mathbb{R}^n$ and $r > 0$, write $B(q,r) := \{y \in \mathbb{R}^n : |y-q| < r\}$ for the open Euclidean ball. We use the following nested coordinate-ball refinement theorem for [smooth manifolds](/page/Smooth%20Manifold), obtained from the [paracompactness](/page/Paracompact%20Space), regularity, and local Euclidean structure of Hausdorff second-countable smooth manifolds: for every open cover $(U_i)_{i \in I}$ of $M$, there are an index set $A$, a map \begin{align*} \iota: A \to I, \end{align*} and, for each $a \in A$, a [coordinate chart](/page/Coordinate%20Chart) \begin{align*} (V_a,\varphi_a), \qquad \varphi_a: V_a \to \varphi_a(V_a) \subset \mathbb{R}^n, \end{align*} a point $c_a \in \varphi_a(V_a)$, and radii $0 < r_a < R_a$ such that, defining \begin{align*} Z_a &:= \varphi_a^{-1}(B(c_a,r_a)),\\ W_a &:= \varphi_a^{-1}(B(c_a,R_a)), \end{align*} the following hold: \begin{align*} \overline{W_a} \subset V_a \cap U_{\iota(a)} \end{align*} for every $a \in A$; the family $(W_a)_{a \in A}$ is locally finite in $M$; and \begin{align*} M = \bigcup_{a \in A} Z_a. \end{align*} Here closures are taken in $M$. This refinement theorem is obtained from [paracompactness](/page/Paracompact%20Space) by first choosing precompact coordinate neighbourhoods subordinate to the given cover and then shrinking each coordinate ball inside the corresponding precompact chart neighbourhood. The nested ball data above is the precise form needed for the cutoff construction in the next step. [guided] The goal is to reduce the arbitrary open cover $(U_i)_{i \in I}$ to coordinate pieces that are small enough to support explicit bump functions. The exact refinement we need is stronger than a bare locally finite refinement: each refining open set must be a coordinate ball, and inside it there must be a smaller concentric coordinate ball whose union still covers $M$. Let $n := \dim M$. For $q \in \mathbb{R}^n$ and $r > 0$, write $B(q,r) := \{y \in \mathbb{R}^n : |y-q| < r\}$ for the open Euclidean ball. We use the nested coordinate-ball refinement theorem for [smooth manifolds](/page/Smooth%20Manifold). Its hypotheses are satisfied because $M$ is assumed Hausdorff, second countable, and locally Euclidean; these assumptions imply [paracompactness](/page/Paracompact%20Space) and local compactness, which are the topological ingredients used to construct locally finite precompact coordinate refinements. Applied to the cover $(U_i)_{i \in I}$, the theorem gives an index set $A$ and a choice map \begin{align*} \iota: A \to I, \end{align*} which records which original open set contains the $a$-th refined neighbourhood. For each $a \in A$, the theorem gives a coordinate chart \begin{align*} (V_a,\varphi_a), \qquad \varphi_a: V_a \to \varphi_a(V_a) \subset \mathbb{R}^n, \end{align*} a point $c_a \in \varphi_a(V_a)$, and radii $0 < r_a < R_a$. We define the smaller and larger coordinate balls by \begin{align*} Z_a &:= \varphi_a^{-1}(B(c_a,r_a)),\\ W_a &:= \varphi_a^{-1}(B(c_a,R_a)). \end{align*} The theorem gives the containment \begin{align*} \overline{W_a} \subset V_a \cap U_{\iota(a)}, \end{align*} where the closure is taken in $M$. It also gives local finiteness of $(W_a)_{a \in A}$, meaning that every point $p \in M$ has an open neighbourhood meeting only finitely many of the sets $W_a$. This local finiteness is essential because it will make all later sums finite near each point. Finally, the smaller coordinate balls still cover $M$: \begin{align*} M = \bigcup_{a \in A} Z_a. \end{align*} This is the only global topological input in the proof; the remaining steps use this nested coordinate-ball data to build smooth cutoffs and normalize them. [/guided] [/step] [step:Construct one smooth bump function on each refining coordinate ball] For each $a \in A$, use the point $c_a \in \varphi_a(V_a) \subset \mathbb{R}^n$ and radii $0 < r_a < R_a$ supplied by the nested coordinate-ball refinement, so that \begin{align*} Z_a = \varphi_a^{-1}(B(c_a,r_a)) \quad\text{and}\quad W_a = \varphi_a^{-1}(B(c_a,R_a)). \end{align*} Define the smooth function \begin{align*} \theta: \mathbb{R} &\to [0,\infty) \\ t &\mapsto \begin{cases} e^{-1/t}, & t > 0,\\ 0, & t \leq 0. \end{cases} \end{align*} For each $a \in A$, define \begin{align*} \beta_a: M &\to [0,\infty) \end{align*} by \begin{align*} \beta_a(p) = \begin{cases} \theta\left(R_a^2 - |\varphi_a(p)-c_a|^2\right), & p \in V_a,\\ 0, & p \in M \setminus \varphi_a^{-1}(B(c_a,R_a)). \end{cases} \end{align*} Because $\theta$ vanishes to infinite order at $0$, this piecewise definition is smooth across the boundary of $\varphi_a^{-1}(B(c_a,R_a))$. Moreover, \begin{align*} \operatorname{supp}\beta_a \subset \overline{W_a} \subset U_{\iota(a)}, \end{align*} and $\beta_a(p) > 0$ for every $p \in Z_a$. [guided] We now build the local smooth functions that will later be normalized. Fix $a \in A$. The previous step did not merely give arbitrary open sets $Z_a \subset W_a$; it gave nested coordinate balls. Thus we already have a point $c_a \in \varphi_a(V_a)$ and radii $0 < r_a < R_a$ such that \begin{align*} Z_a = \varphi_a^{-1}(B(c_a,r_a)) \quad\text{and}\quad W_a = \varphi_a^{-1}(B(c_a,R_a)). \end{align*} Also, \begin{align*} \overline{W_a} \subset V_a \cap U_{\iota(a)}. \end{align*} The standard smooth cutoff on the real line is the function \begin{align*} \theta: \mathbb{R} &\to [0,\infty) \\ t &\mapsto \begin{cases} e^{-1/t}, & t > 0,\\ 0, & t \leq 0. \end{cases} \end{align*} This function is smooth on all of $\mathbb{R}$ and vanishes to infinite order at $t = 0$. The infinite-order vanishing is exactly what makes the extension by zero smooth at the boundary of a ball. Using the coordinate chart $\varphi_a$, define \begin{align*} \beta_a: M &\to [0,\infty) \end{align*} by \begin{align*} \beta_a(p) = \begin{cases} \theta\left(R_a^2 - |\varphi_a(p)-c_a|^2\right), & p \in V_a,\\ 0, & p \in M \setminus \varphi_a^{-1}(B(c_a,R_a)). \end{cases} \end{align*} Inside the coordinate ball $\varphi_a^{-1}(B(c_a,R_a))$, this is a composition of smooth maps: \begin{align*} p \mapsto \varphi_a(p) \mapsto R_a^2 - |\varphi_a(p)-c_a|^2 \mapsto \theta\left(R_a^2 - |\varphi_a(p)-c_a|^2\right). \end{align*} Outside that ball it is zero, and smoothness across the boundary follows because $\theta$ and all of its derivatives vanish at $0$. The [support](/page/Support) is contained in the closure of the larger coordinate ball in $M$: \begin{align*} \operatorname{supp}\beta_a \subset \overline{W_a}. \end{align*} By the refinement construction, \begin{align*} \overline{W_a} \subset U_{\iota(a)}. \end{align*} Thus $\beta_a$ is supported inside the prescribed original member $U_{\iota(a)}$. Also, if $p \in Z_a$, then $\varphi_a(p) \in B(c_a,r_a)$, so \begin{align*} R_a^2 - |\varphi_a(p)-c_a|^2 > R_a^2-r_a^2 > 0. \end{align*} Therefore $\beta_a(p) > 0$ on $Z_a$. [/guided] [/step] [step:Normalize the locally finite bump functions] Define \begin{align*} s: M &\to (0,\infty) \\ p &\mapsto \sum_{a \in A} \beta_a(p). \end{align*} The sum is locally finite because $\operatorname{supp}\beta_a \subset \overline{W_a}$ and the closures of a [locally finite family](/page/Locally%20Finite%20Family) are locally finite. Hence $s$ is smooth. Since the sets $(Z_a)_{a \in A}$ cover $M$ and $\beta_a > 0$ on $Z_a$, for every $p \in M$ at least one summand $\beta_a(p)$ is positive, so $s(p) > 0$. For each $a \in A$, define \begin{align*} \psi_a: M &\to [0,1] \\ p &\mapsto \frac{\beta_a(p)}{s(p)}. \end{align*} Then each $\psi_a$ is smooth, $\operatorname{supp}\psi_a \subset \operatorname{supp}\beta_a \subset U_{\iota(a)}$, the family $(\operatorname{supp}\psi_a)_{a \in A}$ is locally finite because $\operatorname{supp}\psi_a \subset \overline{W_a}$ and the family $(\overline{W_a})_{a \in A}$ is locally finite, and for every $p \in M$, \begin{align*} \sum_{a \in A} \psi_a(p) = \frac{1}{s(p)}\sum_{a \in A}\beta_a(p) = 1. \end{align*} [guided] The bump functions $(\beta_a)_{a \in A}$ are not yet a partition of unity because their sum is not necessarily equal to $1$. We fix this by dividing by their total sum. Define \begin{align*} s: M &\to (0,\infty) \\ p &\mapsto \sum_{a \in A} \beta_a(p). \end{align*} This sum is meaningful pointwise and smooth because it is locally finite. Indeed, $\operatorname{supp}\beta_a \subset \overline{W_a}$, and every point of $M$ has a neighbourhood meeting only finitely many of the closures $\overline{W_a}$, since the closures of a [locally finite family](/page/Locally%20Finite%20Family) are locally finite. On such a neighbourhood, the displayed sum is a finite sum of smooth functions, hence smooth there. Next we verify that division by $s$ is valid. Since $(Z_a)_{a \in A}$ covers $M$, for every $p \in M$ there exists $a \in A$ with $p \in Z_a$. From the previous step, $\beta_a(p) > 0$ on $Z_a$. Therefore \begin{align*} s(p) = \sum_{b \in A}\beta_b(p) \geq \beta_a(p) > 0. \end{align*} Thus $s$ has no zeros. For each $a \in A$, define \begin{align*} \psi_a: M &\to [0,1] \\ p &\mapsto \frac{\beta_a(p)}{s(p)}. \end{align*} Because $\beta_a$ and $s$ are smooth and $s$ is strictly positive, $\psi_a$ is smooth. Its support is contained in the support of $\beta_a$: \begin{align*} \operatorname{supp}\psi_a \subset \operatorname{supp}\beta_a \subset U_{\iota(a)}. \end{align*} The family of supports remains locally finite because $\operatorname{supp}\psi_a \subset \operatorname{supp}\beta_a \subset \overline{W_a}$ and the family $(\overline{W_a})_{a \in A}$ is locally finite. Finally, the normalization gives the partition identity. For every $p \in M$, the relevant sum is finite near $p$, so ordinary algebra applies: \begin{align*} \sum_{a \in A} \psi_a(p) = \sum_{a \in A}\frac{\beta_a(p)}{s(p)} = \frac{1}{s(p)}\sum_{a \in A}\beta_a(p) = 1. \end{align*} Thus $(\psi_a)_{a \in A}$ is a smooth [partition of unity](/page/Partition%20of%20Unity) subordinate to the locally finite refinement. [/guided] [/step] [step:Group the refined partition by the original open cover] For each $i \in I$, define \begin{align*} \rho_i: M &\to [0,1] \\ p &\mapsto \sum_{\substack{a \in A\\ \iota(a)=i}} \psi_a(p). \end{align*} This sum is locally finite because the full family $(\operatorname{supp}\psi_a)_{a \in A}$ is locally finite, as proved using $\operatorname{supp}\psi_a \subset \overline{W_a}$ and local finiteness of $(\overline{W_a})_{a \in A}$. Hence each $\rho_i$ is smooth. Since every $\operatorname{supp}\psi_a$ with $\iota(a)=i$ is contained in $U_i$, and since every subfamily of a locally finite family of closed sets has closed union, we have \begin{align*} \operatorname{supp}\rho_i \subset \bigcup_{\substack{a \in A\\ \iota(a)=i}} \operatorname{supp}\psi_a \subset U_i. \end{align*} The family $(\operatorname{supp}\rho_i)_{i \in I}$ is locally finite because each $\operatorname{supp}\rho_i$ is contained in the union of those locally finite supports $\operatorname{supp}\psi_a$ with $\iota(a)=i$. Finally, for every $p \in M$, \begin{align*} \sum_{i \in I} \rho_i(p) = \sum_{i \in I}\sum_{\substack{a \in A\\ \iota(a)=i}}\psi_a(p) = \sum_{a \in A}\psi_a(p) = 1. \end{align*} Thus $(\rho_i)_{i \in I}$ is a smooth partition of unity subordinate to the original cover $(U_i)_{i \in I}$. [guided] The functions $(\psi_a)_{a \in A}$ are indexed by the refining cover, while the theorem asks for a family indexed by the original cover. The map \begin{align*} \iota: A \to I \end{align*} records which original open set contains the support of each refined function. For each $i \in I$, we therefore collect all refined functions assigned to $i$ and define \begin{align*} \rho_i: M &\to [0,1] \\ p &\mapsto \sum_{\substack{a \in A\\ \iota(a)=i}} \psi_a(p). \end{align*} This is again a locally finite sum. Indeed, near any point $p \in M$, only finitely many functions $\psi_a$ are nonzero, so the sub-sum over those $a$ with $\iota(a)=i$ is finite near $p$. Therefore each $\rho_i$ is smooth. We now check the support condition carefully. If $\iota(a)=i$, then from the construction of $\psi_a$, \begin{align*} \operatorname{supp}\psi_a \subset U_i. \end{align*} The family $(\operatorname{supp}\psi_a)_{a \in A}$ is locally finite, and every subfamily of a locally finite family of closed sets has closed union. Hence the union \begin{align*} \bigcup_{\substack{a \in A\\ \iota(a)=i}} \operatorname{supp}\psi_a \end{align*} is closed in $M$. Since $\rho_i$ vanishes outside this union, its support is contained in that union: \begin{align*} \operatorname{supp}\rho_i \subset \bigcup_{\substack{a \in A\\ \iota(a)=i}} \operatorname{supp}\psi_a \subset U_i. \end{align*} The local finiteness of $(\operatorname{supp}\rho_i)_{i \in I}$ follows from the local finiteness of the refined supports, which in turn follows from $\operatorname{supp}\psi_a \subset \overline{W_a}$ and local finiteness of $(\overline{W_a})_{a \in A}$. Around any point $p \in M$, only finitely many supports $\operatorname{supp}\psi_a$ occur. Therefore only finitely many indices $i=\iota(a)$ can contribute a nonzero $\rho_i$ near $p$. Finally, grouping does not change the total sum. Since all sums are locally finite near each point, we may rearrange them: \begin{align*} \sum_{i \in I} \rho_i(p) = \sum_{i \in I}\sum_{\substack{a \in A\\ \iota(a)=i}}\psi_a(p) = \sum_{a \in A}\psi_a(p) = 1. \end{align*} Thus the family $(\rho_i)_{i \in I}$ satisfies all required properties: smoothness, nonnegativity, local finiteness, support containment in $U_i$, and pointwise sum equal to $1$. [/guided] [/step]