LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A countable coordinate-bump map embeds a manifold in countable Euclidean data

Statement

Let $M^{n}$ be a smooth manifold. Then there are countably many coordinate balls $(U_{j}, x_{j})$ , open sets $V_{j} ⋐ U_{j}$ covering $M$ , and smooth bump functions $ϕ_{j}$ supported in $U_{j}$ and equal to $1$ on $V_{j}$ such that the countable family of blocks

B_{j} (p) : = (ϕ_{j} (p), ϕ_{j} (p) x_{j}^{1} (p), \dots, ϕ_{j} (p) x_{j}^{n} (p)) \in R^{n + 1}

separates points and tangent vectors: if $p \neq q$ , then $B_{j} (p) \neq B_{j} (q)$ for some $j$ , and for each $p \in M$ there is an index $j$ with $p \in V_{j}$ such that the last $n$ coordinates of $B_{j}$ give the chart coordinates on a neighbourhood of $p$ .

Facts & Assumptions

Given: A smooth $n$ -manifold $M$ .

[L1]

Every open cover of $M$ has a countable cover by relatively compact coordinate balls subordinate to it (Every open cover of a manifold has a countable relatively compact coordinate-ball subcover).

[L2]

A chart bump can be chosen with prescribed support inside a chart (A chart bump at a point with prescribed support).

Proof

technique · direct

1.1

Apply [L1] to the trivial cover ${M}$ to obtain countably many relatively compact coordinate balls $U_{j}$ covering $M$ . Shrinking each one slightly inside itself, choose open sets $V_{j} ⋐ U_{j}$ that still cover $M$ . By [L2] there is a smooth bump $ϕ_{j}$ supported in $U_{j}$ and equal to $1$ on $V_{j}$ .

L1L2givenchoose

2.1

Define the coordinate blocks $B_{j}$ as in the statement. Each $B_{j}$ is smooth because it equals the smooth chart-coordinate formula on $U_{j}$ and vanishes off $U_{j}$ .

step 1.1construct

3.1

If $p \neq q$ , choose $j$ with $p \in V_{j}$ . If $B_{j} (p) = B_{j} (q)$ , then $ϕ_{j} (p) = 1$ , hence $ϕ_{j} (q) = 1$ , so both points lie in $U_{j}$ . Equality of the last $n$ coordinates of $B_{j}$ then gives $x_{j} (p) = x_{j} (q)$ , contradicting the injectivity of the chart map. Therefore some block separates $p$ and $q$ .

step 1.1step 2.1algebra

4.1

Fix $p \in M$ and choose $j$ with $p \in V_{j}$ . On $V_{j}$ one has $ϕ_{j} \equiv 1$ , so the last $n$ coordinates of $B_{j}$ are exactly the chart coordinates $x_{j}^{1}, \dots, x_{j}^{n}$ . Their differential is an isomorphism at $p$ , so this single block already detects every nonzero tangent vector at $p$ . Thus the family $(B_{j})$ separates tangent vectors as claimed.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Marco Gualtieri, Topology I: Smooth Manifolds, Part 11 (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 6 (standard reference, not scraped)