Lecture 8: Partitions of Unity
KSM3E04: Analysis on Manifolds — Fall 2025 | September 8, 4:30–6:00 PM
In this lecture, we introduce a very important tool called partitions of unity. This tool lets us take a difficult problem on a complicated set and break it into smaller, easier problems on simple sets. We can then solve each small problem and use the partition of unity to smoothly "glue" the solutions back together to get a global answer. This method is essential for many advanced topics, especially for defining integration on general spaces called manifolds.
1. The Main Idea of a Partition of Unity
Imagine you have a complex shape, let's call it a set $E$. Studying a function over the whole of $E$ at once can be hard. It's often easier to cover $E$ with a collection of simple, overlapping open sets, which we can call $\mathcal{U}$.
A partition of unity is a collection of special smooth functions that helps us focus on one simple set at a time. Think of them as "blending functions". Each blending function is active (non-zero) only on one of the simple sets from our cover $\mathcal{U}$.
These functions have a key property: for any point you pick in your complex shape $E$, when you add up the values of all the blending functions at that point, the sum is exactly 1. This allows us to break down a function $f$ on $E$ into a sum of simpler pieces. Each piece, $f \cdot \varphi_i$, is easy to work with because it is zero everywhere except on a small, simple region. This "divide and conquer" strategy is what makes partitions of unity so powerful.
2. The Existence of Partitions of Unity
It is natural to ask: do such special "blending functions" always exist? The answer is yes. The following theorem guarantees that we can always create a partition of unity for any set $E$ and any open cover $\mathcal{U}$ of that set.
Theorem: The Partition of Unity Theorem
Let $E \subset \mathbb{R}^n$ be any set, and let $\mathcal{U}$ be an open cover of $E$. Then there exists a collection of smooth ($C^\infty$) functions $\Phi = \{\varphi_i\}$, defined on an open set containing $E$, with these four properties:
- Non-negativity: For every point $x \in E$, the value of each function is between 0 and 1. That is, $0 \le \varphi(x) \le 1$ for all $\varphi \in \Phi$.
- Local Finiteness: For any point $x \in E$, you can find a small open neighborhood around it where only a finite number of the functions from $\Phi$ are non-zero. All other functions are zero in that neighborhood.
- Sum to Unity: For any point $x \in E$, the sum of all the functions at that point is exactly 1. That is, $\sum_{\varphi \in \Phi} \varphi(x) = 1$. This sum makes sense because, due to local finiteness, it's always a finite sum.
- Subordination: Each function $\varphi \in \Phi$ is non-zero only within one of the open sets from the cover $\mathcal{U}$. More precisely, for each $\varphi$, there is a set $U \in \mathcal{U}$ such that $\varphi$ is zero everywhere outside of a closed set contained in $U$.
Proof of the Theorem
The proof is built in steps, starting with the simplest case and building up to the most general one.
Step 1: The case where the set $E$ is compact.
When $E$ is compact, the open cover $\mathcal{U}$ has a finite subcover. Let's say $U_1, U_2, \dots, U_m$ are open sets from $\mathcal{U}$ that cover $E$. Our first goal is to "shrink" this open cover to a cover by compact sets whose interiors still cover $E$.
Lemma: Shrinking an Open Cover
Let $E$ be a compact set and $\{U_1, \dots, U_m\}$ be a finite open cover of $E$. Then there exist compact sets $D_i \subset U_i$ for $i=1, \dots, m$ such that $E \subset \bigcup_{i=1}^m \text{interior}(D_i)$.
Click for the proof of the lemma
We construct the sets $D_1, \dots, D_m$ by induction.
First, we define a compact set $C_1 = E \setminus (U_2 \cup \dots \cup U_m)$. This set $C_1$ is closed and bounded, hence compact. Since $\{U_i\}$ covers $E$, we must have $C_1 \subset U_1$. Since $C_1$ is a compact set contained in the open set $U_1$, we can find a compact set $D_1$ such that $C_1 \subset \text{int}(D_1)$ and $D_1 \subset U_1$.
Now, we proceed inductively. Assume for $k > 1$ we have found compact sets $D_1, \dots, D_{k-1}$ with $D_i \subset U_i$. We define the compact set:
By construction, $C_k \subset U_k$. So again, we can find a compact set $D_k$ with $C_k \subset \text{int}(D_k)$ and $D_k \subset U_k$.
After $m$ steps, we have the desired sets $D_1, \dots, D_m$. The collection $\{\text{int}(D_1), \dots, \text{int}(D_m)\}$ covers $E$ because for any $x \in E$, if it is not in $U_2 \cup \dots \cup U_m$, then it is in $C_1 \subset \text{int}(D_1)$. If it is not in $\text{int}(D_1) \cup U_3 \cup \dots \cup U_m$, it is in $C_2 \subset \text{int}(D_2)$, and so on.
With the compact sets $D_i \subset U_i$ constructed, we need smooth functions that are positive on each $D_i$ and zero outside of $U_i$.
Lemma: Existence of Smooth Bump Functions
For any compact set $D$ and any open set $U$ with $D \subset U$, there exists a non-negative smooth function $\psi: \mathbb{R}^n \to [0, \infty)$ which is positive on $D$ and is zero outside of some closed set contained in $U$.
Click for the proof of the lemma
The existence of such functions is a fundamental result in analysis. They are built explicitly.
1. A Basic Smooth Function: The construction starts with the function $f(t) = e^{-1/t^2}$ for $t \neq 0$ and $f(0)=0$. This function is a classic example of a function that is infinitely differentiable ($C^\infty$) everywhere, yet all its derivatives at $t=0$ are zero, making it "flat" at the origin.
2. A Bump on an Interval: Using $f(t)$, we can build a function $g(t) = f(t+1) \cdot f(1-t)$ that is positive only on the open interval $(-1, 1)$ and smoothly goes to zero at the endpoints.
3. A Bump on a Cube: This idea extends to $\mathbb{R}^n$ by defining $G(x_1, \dots, x_n) = g(x_1) g(x_2) \cdots g(x_n)$, which gives a smooth function that is positive on the open cube $(-1,1)^n$ and zero elsewhere. By scaling and translating $G$, we can create a smooth bump function over any cube.
4. A Bump for the Compact Set $D$: Since $D$ is compact and $D \subset U$, we can cover $D$ with a finite number of small open cubes, each of which is contained in $U$. By creating a bump function for each of these cubes and summing them up, we obtain our desired smooth function $\psi$, which is positive on all of $D$ and is zero outside a closed set contained in $U$.
For each pair $(D_i, U_i)$, we use the lemma to find a non-negative smooth function $\psi_i$ that is positive on $D_i$ and has support contained in $U_i$. Since the interiors of the $D_i$ sets cover the compact set $E$, the sum $\Psi(x) = \sum_{i=1}^m \psi_i(x)$ is positive for all $x$ in some open set containing $E$.
On this open set, we can normalize these functions to define our partition of unity:
The collection $\{\varphi_1, \dots, \varphi_m\}$ satisfies all the required properties on $E$.
Step 2: The case where the set $E$ is open.
For an open set $E$, we first write it as an expanding union of compact sets.
Lemma: Exhaustion of an Open Set
Any open set $E \subset \mathbb{R}^n$ can be written as $E = \bigcup_{i=1}^\infty K_i$ where each $K_i$ is compact and $K_i \subset \text{interior}(K_{i+1})$.
Click for the proof of the lemma
Let $E \neq \mathbb{R}^n$. We define the sets $K_i = \{ x \in E \mid \|x\| \le i \text{ and } d(x, \mathbb{R}^n \setminus E) \ge 1/i \}$. Each $K_i$ is closed and bounded, hence compact. One can then show that $K_i \subset \text{int}(K_{i+1})$ and their union is all of $E$. If $E=\mathbb{R}^n$, we can simply use closed balls of radius $i$.
Using this exhaustion, we build our partition of unity by handling the "annular" regions between the compact sets.
Lemma: Creating a Locally Finite Family of Functions
We can construct a locally finite collection of non-negative smooth functions $\{\eta_j\}$ subordinate to the cover $\mathcal{U}$ such that their sum $H(x) = \sum_j \eta_j(x)$ is positive for all $x \in E$.
Click for the proof of the lemma
Let $K_0 = \emptyset$. For each $i \ge 1$, define the compact annulus $A_i = K_i \setminus \text{int}(K_{i-1})$. Also define the open set $W_i = \text{int}(K_{i+1}) \setminus K_{i-2}$. Each $A_i$ is compact and is covered by the open set $W_i$.
For each annulus $A_i$, the collection $\mathcal{U}_i = \{U \cap W_i \mid U \in \mathcal{U}\}$ is an open cover. Since $A_i$ is compact, we can find a finite subcover and, applying the procedure from Step 1, construct a finite partition of unity $\{\psi_{i,k}\}$ on $A_i$, subordinate to $\mathcal{U}_i$. Each $\psi_{i,k}$ can be extended by 0 to be a smooth function on all of $E$.
The full collection of all such functions, $\{\eta_j\} = \bigcup_i \{\psi_{i,k}\}$, is our desired family. It is locally finite because for any point $x \in E$, $x$ is in some $K_m$. The open neighborhood $\text{int}(K_{m+1})$ only intersects a finite number of the annuli $A_i$ (specifically, those with index up to $m+2$), and thus only a finite number of the $\eta_j$ functions are non-zero there. The sum $H(x) = \sum_j \eta_j(x)$ is positive because $x$ is in some $A_i$, where the sum is 1.
Finally, we normalize the locally finite collection $\{\eta_j\}$ to get our partition of unity for the open set $E$:
This collection $\{\varphi_j\}$ is well-defined, smooth, and satisfies all the required properties.
Step 3: The case for an arbitrary set $E$.
For an arbitrary set $E$, we let $U_{total}$ be the union of all open sets in the cover $\mathcal{U}$. This $U_{total}$ is an open set that contains $E$. We can now apply the procedure from Step 2 to this open set. The resulting partition of unity for $U_{total}$ is automatically a valid partition of unity for the subset $E$. This completes the proof.
3. Problems for Practice
Exercise 1: The Patching Problem for Smooth Functions
This problem explores the conditions under which locally defined functions can be "patched together" in a consistent way.
Let $\mathcal{U} = \{U_\alpha\}$ be an open cover of a domain $E \subset \mathbb{R}^2$. Suppose on each non-empty intersection $U_\alpha \cap U_\beta$, we are given a smooth function $g_{\alpha\beta}$ such that the cocycle condition holds: $g_{\alpha\beta} + g_{\beta\gamma} = g_{\alpha\gamma}$ on every triple intersection $U_\alpha \cap U_\beta \cap U_\gamma$. The problem is to determine if we can find a family of functions $\{f_\alpha\}$, where each $f_\alpha$ is smooth on $U_\alpha$, such that $g_{\alpha\beta} = f_\beta - f_\alpha$ on each intersection $U_\alpha \cap U_\beta$.
- Construction on a Contractible Domain: Assume $E = \mathbb{R}^2$. Use a partition of unity $\{\varphi_i\}$ subordinate to a locally finite refinement of $\mathcal{U}$ to explicitly construct the functions $f_\alpha$. This demonstrates that on simple domains, local data can always be patched.
- Topological Obstruction: Now, let $E = \mathbb{R}^2 \setminus \{(0,0)\}$. Consider the open cover given by $U_1 = E \setminus \{(x,0) \mid x \ge 0\}$ and $U_2 = E \setminus \{(x,0) \mid x \le 0\}$. On the intersection $U_1 \cap U_2$, define $g_{12}(z)$ as a smooth branch of the complex argument function $\arg(z)$. Prove that it is impossible to find smooth functions $f_1, f_2$ on their respective domains such that $g_{12} = f_2 - f_1$. Explain precisely where the construction from part 1 fails due to the topology of the punctured plane.
Exercise 2: The Rigidity of Holomorphic Functions
While smooth partitions of unity are abundant in real analysis, their existence fails spectacularly in complex analysis, revealing the rigid nature of holomorphic functions.
Let $E \subset \mathbb{C}$ be a connected open set. A holomorphic partition of unity is a family of holomorphic functions $\{\varphi_i: E \to \mathbb{C}\}$ satisfying the standard properties (locally finite, sum to 1, etc.).
- Let $\{\varphi_i\}$ be a locally finite family of holomorphic functions on $E$ such that $\sum_{i} \varphi_i(z) = 1$ for all $z \in E$. Prove that each function $\varphi_i$ must be a constant.
- Using the result from part 1, prove that the only non-trivial holomorphic partition of unity on a connected open set $E$ is the trivial one consisting of a single function $\varphi_1(z) = 1$. Conclude that it is impossible to construct a holomorphic partition of unity subordinate to any open cover of $E$ that consists of more than one non-empty proper subset.
Exercise 3: A Smooth Function Defining a Closed Set
It is a fundamental result that any closed set in $\mathbb{R}^n$ can be realized as the zero-set of some smooth function. Proving this requires a clever construction using a partition of unity.
Let $A \subset \mathbb{R}^n$ be an arbitrary non-empty closed set. The goal is to prove the existence of a smooth function $F: \mathbb{R}^n \to [0, \infty)$ such that $F(x) = 0$ if and only if $x \in A$.
- Constructing the Right Cover: Consider the open set $U = \mathbb{R}^n \setminus A$. For each integer $k \ge 1$, define the open annulus $U_k = \{x \in U \mid 2^{-k} < d(x,A) < 2^{-k+2}\}$. Show that $\{U_k\}_{k \in \mathbb{Z}}$ forms an open cover of $U$.
- The Gluing Process: Let $\{\varphi_k\}$ be a partition of unity subordinate to this cover $\{U_k\}$. Construct a sequence of smooth "bump" functions $\psi_k$ such that each $\psi_k$ is supported in $U_k$ and attains a maximum value of $2^{-k}$. Define the function $F$ on $U$ by $F(x) = \sum_{k \in \mathbb{Z}} \psi_k(x)$. Explain why this sum is well-defined and smooth on $U$.
- Smoothness at the Boundary: Extend $F$ to all of $\mathbb{R}^n$ by setting $F(x)=0$ for $x \in A$. The major difficulty is to prove that $F$ is smooth at every point $p \in A$. You must show that for any multi-index $\alpha$, the derivative $D^\alpha F(x)$ approaches $0$ as $x \to p$. The careful choice of the cover and the decaying maxima of the functions $\psi_k$ is critical for this part.
Exercise 4: Sketching Whitney's Extension Theorem
Whitney's Extension Theorem provides a remarkable answer to when a function defined on an arbitrary closed set $A \subset \mathbb{R}^n$ can be extended to a $C^k$ function on all of $\mathbb{R}^n$. The key is to specify not just function values on $A$, but a collection of Taylor polynomials (a "jet") that are coherent in a specific way.
Your task is to outline how a partition of unity is the central tool for constructing such an extension. Assume you are given a family of polynomials $\{P_y\}_{y \in A}$ that satisfy the necessary coherence conditions.
- Local Trivial Extensions: For each point $y \in A$, you can define a function $F_y(x) = P_y(x)$ for $x \in \mathbb{R}^n$. This trivially extends the jet at $y$. For each point $z \notin A$, you can find a closest point $y_z \in A$ and define a local extension in a neighborhood of $z$ using the polynomial $P_{y_z}$. This gives a collection of locally defined smooth functions. How do you form an open cover of $\mathbb{R}^n$ from these local definitions?
- The Gluing Process: Explain in detail how to use a partition of unity $\{\varphi_i\}$ subordinate to your cover to blend these local functions ($F_y$ and their counterparts for points outside $A$) into a single global function $F: \mathbb{R}^n \to \mathbb{R}$. Write down the explicit formula for $F(x)$ as a sum involving the functions $\varphi_i$ and the local extensions.
- The Main Difficulty: The crucial and most difficult part of the full proof is showing that the resulting function $F$ is not just smooth, but that its Taylor polynomials on the set $A$ exactly match the originally specified jets $\{P_y\}$. Briefly explain why the properties of a partition of unity (specifically that $\sum \varphi_i = 1$ and the behavior of its derivatives) are essential for this final step to work.
A Look Ahead: Advanced Applications
The following two problems require concepts from differential geometry and the theory of distributions, which may not have been covered yet. They are included to provide a glimpse into the profound importance and power of partitions of unity in building the foundations of modern analysis and geometry. You are encouraged to return to them once you have the necessary background.
Exercise 5: Existence of Riemannian Metrics
- Local Metrics: Let $(U_\alpha, \psi_\alpha)$ be a chart on a manifold $M$. Use the coordinate map $\psi_\alpha: U_\alpha \to \mathbb{R}^n$ to define a local Riemannian metric $g_\alpha$ on $U_\alpha$ by "pulling back" the standard Euclidean inner product from $\mathbb{R}^n$.
- Global Construction: Let $\{\varphi_i\}$ be a partition of unity subordinate to an atlas $\{(U_i, \psi_i)\}$. For each $\varphi_i$, let $g_i$ be the local metric on $U_i$. Define the global tensor $g = \sum_i \varphi_i g_i$. Explain why this results in a smooth symmetric $(0,2)$-tensor field on $M$.
- The Key Property: For any point $p \in M$ and non-zero tangent vector $v \in T_pM$, show that $g_p(v,v) > 0$. Why is it vital that the functions $\varphi_i$ are non-negative for this to hold?
Exercise 6: The Support of a Distribution
The support of a distribution $T$ is the complement of the largest open set on which $T$ is zero. This definition implicitly assumes such a largest set exists. Let $\{U_\alpha\}$ be an arbitrary collection of open sets on which a distribution $T$ is zero. Prove that $T$ is also zero on their union $U = \bigcup_\alpha U_\alpha$, thus justifying the definition.