Atlas Belongs to Unique Differentiable Structure
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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 $\AA \in \FF$.
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.