CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck 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 $U$ .

[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

1.1F1choose

Compactness gives a finite subcover $U_{0} = {U_{1}, \dots, U_{n}}$ of $U$ .

2.1L1L2step 1.1choose

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.

3.1L3L4step 2.1construct

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 $h_{1} + \dots + h_{n} = 1$ .

4.1step 3.1∎

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

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources