TheoremStatement: 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 paracompact Hausdorff space admits a locally finite subordinate partition of unity

Statement

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

Facts & Assumptions

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

[L1]

There are locally finite covers ${V_{s}}$ , ${W_{s}}$ and $U_{s} \in U$ with $\overline{V_{s}} \subseteq W_{s} \subseteq \overline{W_{s}} \subseteq U_{s}$ (Under choice, every open cover of a paracompact Hausdorff space has locally finite open refinements ${V_{s}}$ and ${W_{s}}$ with $\overline{V_{s}} \subseteq W_{s} \subseteq \overline{W_{s}} \subseteq U_{s}$ ).

[L2]

Every paracompact Hausdorff space is normal (Every paracompact Hausdorff space is normal).

[L3]

Under dependent choice, Urysohn's lemma separates disjoint closed sets in a normal space by a continuous map into $[0, 1]$ (Urysohn's lemma, under the axiom of dependent choice: in a normal space two disjoint closed sets are separated by a continuous function into $[0, 1]$ , and conversely such a space is normal, clause 1).

[L4]

If ${f_{s} : X \to [0, \infty)}_{s \in S}$ is a continuous family with locally finite cozero family and everywhere-positive sum $f = \sum_{s} f_{s}$ , then the functions $φ_{s} = f_{s} / f$ form a partition of unity, with the same cozero sets and supports as the corresponding $f_{s}$ (A locally finite nonnegative family with positive pointwise sum normalizes to a partition of unity).

Proof

technique · constructive

1.1

Apply [L1] to obtain $V_{s}, W_{s}, U_{s}$ as stated.

L1construct

2.1

By [L2], $X$ is normal. For each $s$ , the closed sets $\overline{V_{s}}$ and $X ∖ W_{s}$ are disjoint, so [L3] gives a continuous $f_{s} : X \to [0, 1]$ equal to $1$ on $\overline{V_{s}}$ and $0$ on $X ∖ W_{s}$ .

step 1.1L2L3choose

3.1

The cozero set of $f_{s}$ lies in $W_{s}$ , while its support lies in $\overline{W_{s}} \subseteq U_{s}$ ; since ${W_{s}}$ is locally finite, so is the cozero family.

step 1.1step 2.1

3.2

Because ${V_{s}}$ covers $X$ and $f_{s} = 1$ on $V_{s}$ , the pointwise sum $\sum_{s} f_{s}$ is positive everywhere.

step 1.1step 2.1

4.1

By [L4], the normalized functions $φ_{s} = f_{s} / (\sum_{t} f_{t})$ form a locally finite partition of unity; their supports equal those of $f_{s}$ , so step 3.1 makes the partition subordinate to $U$ .

step 3.1step 3.2L4discharge-construct∎

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Used by

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Sources

Dartmouth Point-Set Topology, Lecture 25 (standard reference, not scraped)
J. Robbin, Partitions of Unity (standard reference, not scraped)
Topology 262 notes (California State University, Northridge) (standard reference, not scraped)
R. Gardner, Notes on Munkres Section 41: Paracompactness (East Tennessee State University) (standard reference, not scraped)