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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

1.1L1construct

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

2.1step 1.1L2L3choose

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 $X ∖ W_{s}$ .

3.1step 1.1step 2.1

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.

3.2step 1.1step 2.1

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

4.1step 3.1step 3.2L4discharge-construct∎

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 $U$ .

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources