# Atlas Belongs to Unique Differentiable Structure

Jump to navigation
Jump to search

## Theorem

Let $M$ be a locally Euclidean space of dimension $d$.

Let $\AA$ be an atlas on $M$.

Then there exists a unique differentiable structure $\FF$ on $M$ with $\AA \in \FF$.

## Proof

Let $\FF$ be the equivalence class of $\AA$ under the relation of compatibility.

By Compatibility of Atlases is Equivalence Relation, this is indeed an equivalence relation.

By definition we have $\AA \in \FF$.

By Relation Partitions Set iff Equivalence, $\FF$ is an element of the partition of equivalence classes.

By definition, the elements of a partition are pairwise disjoint.

Therefore if $\GG \ne \FF$ is an element of the partition, we must have:

- $\AA \notin \GG$

Therefore $\AA$ belongs to exactly one differentiable structure.

$\blacksquare$

This article, or a section of it, needs explaining.Clarification is needed as to why this result should be categorised in Category:Manifolds.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |