Uniqueness of Coordinates

Statements

Theorem. Let the manifold $M$ be a hausdorff space with a countable basis of open sets. If $\mathcal{V}=\{V_{\beta },{\psi }_{\beta }{\}}_{\beta \in \mathbb{N}}$ is a cover of $M$ by ${\mathcal{C}}^{\infty }$-compatible coordinate neighborhoods, then there exists a unique ${\mathcal{C}}^{\infty }$ structure on $M$ containing these coordinate neighborhoods.

References

@Boo75 Chapter 3 Theorem 1.3