ExampleConstruction: AI-generatedVerification: Not suppliedjudge 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.

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Under choice and dependent choice, a finite subordinate partition of unity for a two-set cover of a compact interval

Example

On $[0, 1]$ , take the open-in-the-subspace cover $U = (- 1, \frac{3}{4}) \cap [0, 1]$ and $V = (\frac{1}{4}, 2) \cap [0, 1]$ . Define $φ (x) = \max {0, \min {1, 2 - 3 x}}, ψ (x) = 1 - φ (x) .$ Then $φ$ and $ψ$ are continuous, nonnegative, and sum to one. Their supports are contained respectively in $[0, \frac{2}{3}] \subseteq U$ and $[\frac{1}{3}, 1] \subseteq V$ , so they are a finite subordinate partition of unity.

The explicit pair is an instance of the existence theorem Under choice and dependent choice, every open cover of a compact Hausdorff space admits a finite subordinate partition of unity, under its stated Choice and Dependent Choice hypotheses.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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)