PropositionStatement: Literature-sourcedProof: 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.

Every compact space is paracompact

Statement

Every compact topological space is paracompact.

Facts & Assumptions

Given: A compact topological space $X$ and an open cover $U$ of $X$ .

[F1]

Compactness means that $U$ has a finite subcover (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

[F2]

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

Proof

technique · direct

1.1

By compactness, fix a finite subfamily $V \subseteq U$ covering $X$ .

F1choose

2.1

The family $V$ is open, covers $X$ , refines $U$ , and is locally finite because every point has the neighbourhood $X$ , which meets only members of the finite family $V$ .

step 1.1

3.1

Thus $V$ is the refinement required by [F2], and $X$ is paracompact.

F2step 2.1∎

Depends on

Used by

Under choice and dependent choice, every open cover of a compact Hausdorff space admits a finite subordinate partition of unity Corollary
Assuming choice, paracompactness is not open-hereditary: ω₁ inside ω₁+1 Counterexample
Assuming choice, refuted: paracompactness is hereditary False statement
Refuted: every paracompact space is normal False statement
The tautological degree-one class is well defined and fiber generating Lemma

Dependency tree · two levels

8 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

Dartmouth Point-Set Topology, Lecture 25 (standard reference, not scraped)
R. Gardner, Notes on Munkres Section 41: Paracompactness (East Tennessee State University) (standard reference, not scraped)