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

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For a Hausdorff space, paracompactness is equivalent, under choice and dependent choice, to the existence of a locally finite subordinate partition of unity for every open cover

Statement

Assume the Axiom of Choice and the Axiom of Dependent Choice. For a Hausdorff space $X$ , the following are equivalent: $X$ is paracompact; every open cover of $X$ admits a locally finite partition of unity subordinate to it.

Facts & Assumptions

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

[L1]

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

[F1]

In a subordinate partition, cozero sets are open, form a locally finite family, and each support lies in a member of $\mathcal U$ (Locally finite partitions of unity and subordination to an open cover, Zero sets and cozero sets of continuous real-valued functions).

[F2]

Paracompactness asks for a locally finite open refining cover (Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word).

Proof

technique · direct

1.1

If $X$ is paracompact, [L1] supplies the asserted partition for $\mathcal U$ .

1.2

Conversely, suppose every open cover admits such a partition. For the partition subordinate to $\mathcal U$ , the cozero sets cover $X$ because their functions sum to one.

1.3

Each cozero set is open, locally finite among the cozero family, and contained in its support and hence in a member of $\mathcal U$ ; it is therefore a locally finite open refinement of $\mathcal U$ .

2.1

By [F2], step 1.3 proves that $X$ is paracompact, completing the equivalence.

F2step 1.1step 1.3∎

Depends on

Used by

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Sources

J. Robbin, Partitions of Unity (standard reference, not scraped)