CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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Under choice and dependent choice, every open cover of a compact Hausdorff space admits a finite subordinate partition of unity

Statement

Assume the Axiom of Choice and the Axiom of Dependent Choice. Every open cover of a compact Hausdorff space admits a finite partition of unity subordinate to that cover.

Facts & Assumptions

Given: Choice, dependent choice, a compact Hausdorff space $X$ , and an open cover $\mathcal U$ .

[L1]

A compact space is paracompact (Every compact space is paracompact).

[L2]

A paracompact Hausdorff space has a locally finite partition subordinate to each of its open covers (Under choice and dependent choice, every open cover of a paracompact Hausdorff space admits a locally finite subordinate partition of unity).

[L3]

A locally finite sum of continuous nonnegative functions is continuous (A locally finite family of continuous nonnegative functions has a continuous pointwise sum).

[L4]

Closure commutes with a locally finite union (Locally finite families remain locally finite after taking closures, closure commutes with their union, and a locally finite union of closed sets is closed).

[F1]

Compactness gives a finite subcover (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Proof

technique · direct

1.1

Compactness gives a finite subcover $\mathcal U_0=\{U_1,\ldots,U_n\}$ of $\mathcal U$ .

F1choose

2.1

By [L1] and [L2], apply the partition theorem to the finite cover $\mathcal U_0$ and take a locally finite partition $\{\varphi_s\}_{s\in S}$ subordinate to it.

L1L2step 1.1choose

3.1

Assign each $\varphi_s$ to the first $U_j$ containing its support, and set $h_j$ equal to the corresponding sum. By [L3] the $h_j$ are continuous; by [L4] their supports are contained in $U_j$ ; and $h_1+\cdots+h_n=1$ .

L3L4step 2.1construct

4.1

Discarding the zero $h_j$ leaves a finite subordinate partition of unity.

step 3.1∎

Depends on

Used by

Under choice and dependent choice, a finite subordinate partition of unity for a two-set cover of a compact interval Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 78 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Robbin, Partitions of Unity (standard reference, not scraped)
Topology 262 notes (California State University, Northridge) (standard reference, not scraped)
General Topology notes (University of Göttingen) (standard reference, not scraped)