# Uniqueness of the smooth manifold structure in the Smooth Manifold Chart Lemma, Lee's book.

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.