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

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A countable coordinate-ball cover has a countable locally finite shrinking

Statement

Assume countable choice ${A C}_{ω}$ . Let $(U_{n})_{n \geq 1}$ be a countable cover of a smooth manifold $M$ by coordinate balls with compact closures. Then there is an at-most-countable index set $I$ and families of open sets $(W_{k})_{k \in I}$ and coordinate balls $(V_{k})_{k \in I}$ such that $M = ⋃_{k \in I} W_{k}$ , each $\overline{W_{k}} \subseteq V_{k} \subseteq \overline{V_{k}} \subseteq U_{n (k)}$ for some index $n (k)$ , and the family $(V_{k})$ is locally finite. The index set may be finite or empty.

Facts & Assumptions

Given: Countable choice and a countable cover $(U_{n})_{n \geq 1}$ of $M$ by coordinate balls with compact closures.

[L1]

Coordinate balls form a basis of the underlying topological manifold, and the basis balls supplied there have compact closures (Coordinate balls form a basis of a topological manifold).

[L2]

In a regular space, if $x \in U$ with $U$ open, then there is an open set $V$ with $x \in V \subseteq \overline{V} \subseteq U$ (A space is regular if and only if every point has a neighbourhood base of closed neighbourhoods, if and only if $x \in U$ open gives an open $V$ with $x \in V \subseteq \overline{V} \subseteq U$ ).

[L3]

Compact subsets of Hausdorff spaces are closed, and closed subspaces of compact spaces are compact (In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact).

[A1]

Smooth manifolds are Hausdorff and regular.

[A2]

Countable choice selects one finite covering list for each compact annulus from independently specified nonempty sets of lists (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenL3

Put $H_{r} : = ⋃_{i = 1}^{r} \overline{U_{i}}$ for $r \geq 1$ . Each $H_{r}$ is compact, and the interiors of the $H_{r}$ cover $M$ because $U_{r} \subseteq int (H_{r})$ .

2.1step 1.1choose

Set $r_{1} = 1$ . Recursively, compactness of $H_{r_{m}}$ and the open cover $(int (H_{r}))_{r \geq 1}$ give an integer $r_{m + 1} > r_{m}$ such that $H_{r_{m}} \subseteq int (H_{r_{m + 1}})$ . Put $K_{m} : = H_{r_{m}}$ and $K_{- 1} = K_{0} = \emptyset$ . Then $K_{m} \subseteq int (K_{m + 1})$ and the interiors of the $K_{m}$ cover $M$ .

3.1L1L2L3step 2.1construct

Put $A_{1} : = K_{1}$ and $A_{m} : = K_{m} ∖ int (K_{m - 1})$ for $m \geq 2$ ; each $A_{m}$ is compact by [L3]. For each $m$ , consider all tuples $(n, W, V)$ with $V$ a coordinate ball, $W$ open, and $\overline{W} \subseteq V \subseteq \overline{V} \subseteq U_{n} \cap int (K_{m + 1}) ∖ K_{m - 2} .$ Their $W$ -sets cover $A_{m}$ : for an individual $x \in A_{m}$ , take any $U_{n}$ containing $x$ and successively use [L2] and [L1] inside the displayed open neighbourhood to construct such $V$ and $W$ around $x$ . This proves existence at each point without a simultaneous point-indexed choice. By compactness, the set of finite ordered lists of these tuples whose $W$ -sets cover $A_{m}$ is nonempty (and contains the empty list when $A_{m} = \emptyset$ ).

4.1A2step 2.1step 3.1choose

Use [A2] once on the countable family of nonempty finite-list sets from step 3.1. Concatenate the chosen lists into an at-most-countable indexed family $(W_{k}, V_{k})_{k \in I}$ ; take $I = \emptyset$ when every list is empty. They cover $M$ because the annuli $A_{m}$ cover $M$ . Given $y \in M$ , choose $m$ with $y \in int (K_{m})$ . If $V_{k}$ came from annulus $A_{j}$ with $j \geq m + 3$ , then $V_{k} \cap int (K_{m}) = \emptyset$ because $K_{j - 2} \supseteq K_{m}$ . Only finitely many tuples came from the remaining finitely many annuli, so $(V_{k})$ is locally finite.

5.1step 4.1∎

Hence $(W_{k})$ and $(V_{k})$ give the required countable locally finite shrinking.

Depends on

Used by

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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)