LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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

Statement

Let $(U_{n})_{n \geq 1}$ be a countable cover of a smooth manifold $M$ by coordinate balls with compact closures. Then there are countable families of open sets $(W_{k})_{k \geq 1}$ and coordinate balls $(V_{k})_{k \geq 1}$ such that $M = ⋃_{k} W_{k}$ , 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.

Facts & Assumptions

Given: 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.

Proof

technique · direct

1.1

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 $U_{r} \subseteq int (H_{r})$ .

givenL3

2.1

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 $M$ .

step 1.1choose

3.1

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]. Fix $x \in A_{m}$ , choose some $U_{n}$ containing $x$ , and set $O_{x} : = U_{n} \cap int (K_{m + 1}) ∖ K_{m - 2} .$ By step 2.1, this is an open neighbourhood of $x$ . Apply [L2] to choose an open $R_{x}$ with $x \in R_{x} \subseteq \overline{R_{x}} \subseteq O_{x}$ , and then [L1] to choose a coordinate ball $V_{x}$ with $x \in V_{x} \subseteq R_{x}$ and compact closure. Thus $\overline{V_{x}} \subseteq \overline{R_{x}} \subseteq O_{x}$ . Applying [L2] inside $V_{x}$ , choose an open set $W_{x}$ with $x \in W_{x} \subseteq \overline{W_{x}} \subseteq V_{x} .$ Compactness of $A_{m}$ gives finitely many such pairs covering $A_{m}$ .

L1L2L3step 2.1choose

4.1

Collect the finitely many pairs from each $A_{m}$ into sequences $(W_{k})$ and $(V_{k})$ . They are countable, and they cover $M$ because the annuli $A_{m}$ cover $M$ . Given $y \in M$ , choose $m$ with $y \in int (K_{m})$ . If $V_{k}$ is attached to $A_{j}$ with $j \geq m + 3$ , then step 3.1 gives $V_{k} \cap int (K_{m}) = \emptyset$ because $K_{j - 2} \supseteq K_{m}$ . Hence the neighbourhood $int (K_{m})$ meets only the finitely many families attached to $A_{1}, \dots, A_{m + 2}$ . Thus $(V_{k})$ is locally finite.

step 2.1step 3.1algebra

5.1

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

step 4.1∎

Depends on

Used by

Dependency tree · two levels

19 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)