Given a manifold sub-atlas is the differentiable structure unique?

Question

If we are given a subatlas $\{ (U_\alpha,\phi_\alpha)\}_\alpha$ of a manifold $\mathcal M$, is the differentiable structure unique (i.e. is its maximal extension unique)?

Answer 1

Given any sub-atlas $S$, you may obtain a maximal atlas $A$ containing it by adjoining all compatible charts. Now suppose $A'$ is another maximal atlas containing $S$. Then in particular all the charts in $A'$ are compatible with $S$, so by construction of $A$ we must have $A' \subset A$. Since we assumed both to be maximal, we conclude that $A' = A$.