TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-29

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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

1.1F1F2F3

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.

2.1F1L1L2step 1.1

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.

2.2F1L3A1step 1.1

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.

3.1step 1.1step 2.1step 2.2∎

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.

Topological manifolds are metrizable and paracompact

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources