TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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Topological manifolds are sigma-compact

Statement

Every topological manifold is $σ$ -compact.

Facts & Assumptions

Given: A topological manifold $M$ .

[F1]

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

[F2]

Coordinate balls form a basis of the topology, and each coordinate ball has compact closure in $M$ (Coordinate balls form a basis of a topological manifold).

Proof

technique · direct

1.1

By [F1] choose a countable basis $B = (B_{n})_{n \in N}$ for the topology of $M$ . By [F2], for each $n$ and each point of $B_{n}$ there is a coordinate ball contained in $B_{n}$ whose closure is compact. Replacing each $B_{n}$ by all coordinate balls it contains, we obtain a countable basis $(U_{k})_{k \in N}$ of coordinate balls with compact closures.

F1F2choose

2.1

Every $\overline{U_{k}}$ is compact by [F2], and the family $(\overline{U_{k}})_{k \in N}$ covers $M$ because the basis $(U_{k})$ does.

F2step 1.1

$M = ⋃_{k \in N} \overline{U_{k}}$

is a countable union of compact subsets.

3.1

Therefore $M$ is $σ$ -compact.

step 2.1∎

Depends on

Used by

Topological manifolds are metrizable and paracompact Theorem

Dependency tree · two levels

7 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)