LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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Every manifold has a compact exhaustion

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Every smooth manifold admits a compact exhaustion.

Facts & Assumptions

Given: ${A C}_{ω}$ and a smooth manifold $M$ .

[L1]

Under ${A C}_{ω}$ , 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). This is the only use of countable choice; the finite indices below are chosen canonically as least eligible integers.

[A1]

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

Proof

technique · direct

1.1L1A1givenconstruct

If $M = \emptyset$ , the constant sequence $K_{k} = \emptyset$ is already a compact exhaustion. Otherwise take the countable cover $U_{1}, U_{2}, \dots$ of [L1], repeating a ball if the supplied cover is finite. Let $m_{1}$ be the least integer with $\overline{U_{1}} \subseteq ⋃_{i \leq m_{1}} U_{i}$ . Given $m_{k}$ , let $m_{k + 1} > m_{k}$ be the least integer such that the compact set $⋃_{i \leq m_{k}} \overline{U_{i}}$ lies in $⋃_{i \leq m_{k + 1}} U_{i}$ . Such integers exist by [A1], and selecting their least values uses no further choice.

2.1step 1.1

In the nonempty case 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 $M$ .

3.1step 1.1step 2.1∎

The empty case of step 1.1 and the nonempty case of step 2.1 both give a compact exhaustion in the sense of Compact exhaustions of a manifold.

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Used by

Dependency tree · two levels

12 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)