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A partition of unity replaces a cover by continuous non-negative weights that add to one. Subordination keeps each weight inside an allowed open set; local finiteness ensures only finitely many weights are active near any point. These simple-looking conditions let local data be blended globally. They also expose a common trap: a weighted sum of local distances need not be a metric unless separation and compatibility are proved.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Define a partition of unity
A family {φi} of continuous functions X→[0,1] is a partition of unity when the family of supports is locally finite and Σiφi(x)=1 for every x.
The support is the closure of the set where the function is non-zero. Using only the non-zero set can miss boundary points that matter for subordination.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Make the partition subordinate to a cover
Given an open cover {Ui}, a partition is subordinate when each support supp φi lies inside an assigned cover member. The index sets need not match exactly, but the assignment must be explicit.
Paracompact Hausdorff spaces admit subordinate partitions of unity for open covers. Separation conventions vary by text, so the Hausdorff condition should not be hidden.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Build a two-function partition on the line
Cover R by U1=(−∞,1) and U2=(−1,∞). Define φ1=1 for x≤−1/2, φ1=(1/2)−x for −1/2≤x≤1/2, and φ1=0 for x≥1/2. Let φ2=1−φ1.
| x | φ1(x) | φ2(x) | Sum |
|---|---|---|---|
| −1 | 1 | 0 | 1 |
| −1/2 | 1 | 0 | 1 |
| 0 | 1/2 | 1/2 | 1 |
| 1/2 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 |
The support of φ1 is (−∞,1/2], contained in U1; the support of φ2 is [−1/2,∞), contained in U2. The buffer prevents the supports from touching the forbidden cover boundaries.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. See why local finiteness matters
A family is locally finite when every point has a neighbourhood meeting only finitely many supports. Then a sum such as Σiφi(x)fi(x) is locally a finite sum of continuous functions and is therefore continuous.
| Family property | Pointwise finite | Locally finite |
|---|---|---|
| Meaning | Finitely many terms non-zero at each point | One neighbourhood meets only finitely many supports |
| Enough for continuity of arbitrary sums? | Not by itself | Yes for locally finite continuous sums |
| Controls nearby behaviour? | Weakly | Directly |
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Glue scalar and vector-valued data
If fi is a local scalar or vector-valued construction on Ui, then F=Σiφifi blends the local data wherever multiplication and addition make sense. Subordination ensures each term is used only where its local definition is valid.
The blend may smooth transitions, but it does not preserve every nonlinear property. A weighted average of invertible matrices, probability models or solutions to nonlinear equations can leave the desired class.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Try a naive metric formula
Suppose Ui carries a local metric di. A tempting formula is D(x,y)=Σiφi(x)φi(y)min(1,di(x,y)), with terms used only when both points lie in Ui.
This expression is symmetric and non-negative, but separation and the triangle inequality are not automatic. If no common active chart distinguishes x and y, the sum can be zero for distinct points; varying weights can also disrupt metric inequalities.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. State the genuine metrisation route
A paracompact Hausdorff locally metrizable space is metrizable. Standard proofs use locally finite refinements, partitions of unity and carefully designed functions or pseudometrics that collectively separate points.
The global metric is not obtained by writing one plausible sum and hoping. Each local ingredient, extension, compatibility condition and separation step has to be verified.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the established cover routes
Use star refinements and paracompactness for the cover control that precedes a subordinate partition. Use Nagata-Smirnov metrisation for a σ-locally finite base criterion.
The guides to Bing metrisation and Moore metrisation show alternative ways that discrete families, developments and separation hypotheses produce a global metric.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
A partition of unity is not a disjoint partition of the space. Several weights may be positive at once; their overlap is exactly what permits a continuous transition.
Local finiteness is stronger than pointwise finiteness, and local metrizability is weaker than global metrizability. A partition can glue linear data without automatically gluing a nonlinear invariant.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use a proof and source checklist
State the cover, separation assumptions, support convention, local finiteness, subordination map, normalisation, continuity of sums, local objects, extension rule and the proof that the global construction retains its claimed property.
The University of Wisconsin note Partitions of Unity states the paracompact Hausdorff characterisation. Purdue’s Lecture Notes on K-Theory develops locally finite supports and subordinate partitions as the mechanism for global constructions.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why take the closure in the support?
Because continuity can force non-zero values arbitrarily near a boundary point, and subordination must control that boundary as well.
Can a compact space use a finite partition?
Yes. A finite subcover and the standard construction give a finite subordinate partition under the usual Hausdorff assumptions.
Does every open cover have a partition of unity?
Not in every space. Paracompactness with the stated separation assumptions is the key global condition.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a changed-buffer check
Replace the transition endpoints −1/2 and 1/2 by −a and a. Determine which 0 Continue through the Topology root, the Gold Standard of Education and the How X Works hub. Top · Contents · Previous · How X Works