LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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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.
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Every manifold has a compact exhaustion

Statement

Every smooth manifold admits a compact exhaustion.

Facts & Assumptions

Given: A smooth manifold $M$ .

[L1]

The manifold has a countable cover by relatively compact coordinate balls $U_{1}, U_{2}, \dots$ (Every open cover of a manifold has a countable relatively compact coordinate-ball subcover).

[A1]

A compact subset of a space covered by open sets has a finite subcover.

Proof

technique · direct

1.1

Start from the countable cover $U_{1}, U_{2}, \dots$ of [L1]. Recursively choose integers $1 \leq m_{1} < m_{2} < \dots$ such that $\overline{U_{1}} \subseteq ⋃_{i \leq m_{1}} U_{i}$ and, for each $k \geq 1$ , the compact set $⋃_{i \leq m_{k}} \overline{U_{i}}$ is contained in $⋃_{i \leq m_{k + 1}} U_{i}$ ; this is possible by [A1].

L1A1choose

2.1

Put $K_{k} : = ⋃_{i \leq m_{k}} \overline{U_{i}}$ . Each $K_{k}$ is compact, step 1.1 gives $K_{k} \subseteq int (K_{k + 1})$ , and every point of $M$ lies in some $K_{k}$ because the $U_{i}$ cover $M$ .

step 1.1

3.1

Therefore $(K_{k})$ is a compact exhaustion in the sense of Compact exhaustions of a manifold.

step 2.1∎

Depends on

Used by

Every smooth manifold admits a smooth proper exhaustion function Theorem

Dependency tree · two levels

6 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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)