TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29‡ rests on unproved material (inherited)

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.

‡ Rests on 1 statement not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Under choice, a regular T₁ space with a σ-locally-finite basis has a compatible normal sequence. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Topological manifolds are metrizable and paracompact

Statement

Assume the choice principles carried by the cited topology results: ${A C}_{ω}$ for the Lindelof step and the Axiom of Choice for the metrization corollary. Then every topological manifold is regular, metrizable, and paracompact.

Facts & Assumptions

Given: A topological manifold $M$ , together with the choice hypotheses named in the Statement.

[F1]

A topological manifold is Hausdorff and second countable (Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces).

[F2]

A topological manifold is locally compact (Topological manifolds are locally compact and locally path connected).

[F3]

In a locally compact Hausdorff space, every point has an open neighbourhood with compact closure inside any given open neighbourhood; in particular such a space is regular (In a locally compact Hausdorff space every point has a neighbourhood base of compact sets, every open subspace and every closed subspace is locally compact, every open set around a point contains an open set with compact closure inside it, and every compact set sits inside an open set with compact closure).

[L1]

Assuming ${A C}_{ω}$ , every second-countable space is Lindelof (Assuming countable choice, every second countable space is Lindelöf).

[L2]

Assuming ${A C}_{ω}$ , every regular Lindelof space is paracompact (Under countable choice, every regular Lindelöf space is paracompact).

[L3]

Assuming the Axiom of Choice, every regular $T_{1}$ second-countable space is metrizable (Under choice, every regular $T_{1}$ second-countable space is metrizable).

[A1]

Every Hausdorff space is $T_{1}$ .

Proof

technique · direct

1.1

By [F1] the manifold $M$ is Hausdorff and second countable, and by [F2] it is locally compact. Therefore [F3] applies and shows that $M$ is regular.

F1F2F3

2.1

The second-countability hypothesis from [F1] and the declared ${A C}_{ω}$ assumption let us apply [L1], so $M$ is Lindelof. Then [L2] applies to the regular space of step 1.1 and yields paracompactness.

F1L1L2step 1.1

2.2

By [A1], the Hausdorff property in [F1] implies $T_{1}$ . Hence [L3] applies to the regular, $T_{1}$ , second-countable space $M$ and yields metrizability.

F1L3A1step 1.1

3.1

Step 1.1 proves regularity, step 2.1 proves paracompactness, and step 2.2 proves metrizability. The theorem Topological manifolds are sigma-compact is recorded in the dependency closure because it is another global consequence of the same convention, though it is not needed in the chosen proof route here.

step 1.1step 2.1step 2.2∎

Depends on

Used by

Manifold conventions and the role of second countability Remark

Dependency tree · two levels

30 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

Rob van der Vorst, Introduction to differentiable manifolds, §1, Theorem 1.4 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.2 (standard reference, not scraped)