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Here there is a description of the Lemma I'm referring to, and I will use the same notation.

The Uniqueness Part of the Smooth-Manifold-Chart-Lemma in John M. Lee's Introduction to Smooth Manifolds.

I want to be sure I understood the uniqueness assertion of this Lemma.

Here is my proof.

Let $\mathcal{V}$ be the topology on $M$ constructed in the proof of that Lemma, with $\mathcal{B}$ its basis (also given in the proof of that Lemma). Let $\mathcal{N}$ be a topology on $M$ for which $\{(U_\alpha,\phi_\alpha)\}$ is a smooth atlas for $M$.

Then for each $\alpha$ I have that $\phi_\alpha:(U_\alpha,\mathcal{N}) \to \mathbb{R}^n$ is contunuous, injective and open. If $V$ is any open subset of $\mathbb{R}^n$, then by the above continuity I have $\phi_\alpha^{-1}(V)$ open in $(U_\alpha,\mathcal{N})$, hence open in $(M,\mathcal{N})$ so $\mathcal{B}\subseteq \mathcal{N}$ and so $\mathcal{V}\subseteq \mathcal{N}$.

Conversely let $A\in \mathcal{N}$. Then $A\cap U_\alpha$ is open in $(U_\alpha,\mathcal{N})$, so $\phi_\alpha(A \cap U_\alpha)$ is open in $\mathbb{R}^n$. Then $A \cap U_\alpha=\phi_\alpha^{-1}(\phi_\alpha(A \cap U_\alpha))$ is an element of $\mathcal{B}$, so $A\cap U_\alpha$ is in $\mathcal{V}$. So $A=\bigcup_\alpha(A \cap U_\alpha) \in \mathcal{V}$.

So $\mathcal{N}=\mathcal{V}$ $\quad$ Q.E.D.

The uniqueness of the smooth structure follows from the fact that on a topological manifold every smooth atlas determines a unique smooth structure by Proposition 1.17 of that book.

Sorry for asking an already asked question but I want to be sure I got the point.

The question is if my proof of the uniqueness of the topology is correct.

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