CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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 $U$ .

[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 $U_{0} = {U_{1}, \dots, U_{n}}$ of $U$ .

F1choose

2.1

By [L1] and [L2], apply the partition theorem to the finite cover $U_{0}$ and take a locally finite partition ${φ_{s}}_{s \in S}$ subordinate to it.

L1L2step 1.1choose

3.1

Assign each $φ_{s}$ 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 $h_{1} + \dots + h_{n} = 1$ .

L3L4step 2.1construct

4.1

Discarding the zero $h_{j}$ leaves a finite subordinate partition of unity.

step 3.1∎

Depends on

Used by

Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)