I am trying to understand the proof of Lemma 1.35 (Smooth Manifold Chart Lemma) of John. M. Lee's Introduction to Smooth Manifolds, 2nd Edition.
The Lemma is an existence-and-uniqueness-lemma. I understand the existence part of it but not the uniqueness part. Here I state the Lemma and the proof of the existence part (the proof is essentially just a detailed version of the proof given in Lee's book.)
LEMMA. Let $M$ be a set and $\{U_\alpha\}_{\alpha\in J}$ be a collection of subsets of $M$, along with maps $\varphi_\alpha:U_\alpha\to\mathbf R^n$, such that the following properties are satisfied:
(i) $\forall \alpha\in J$: $\varphi_\alpha$ is an injective map and $\varphi_\alpha(U_\alpha)$ is open in $\mathbf R^n$.
(ii) $\forall \alpha,\beta\in J$: the sets $\varphi_\alpha(U_\alpha\cap U_\beta)$ and $\varphi_\beta(U_\alpha\cap U_\beta)$ are open in $\mathbf R^n$.
(iii) $\forall\alpha,\beta\in J$: $U_\alpha\cap U_\beta\neq \emptyset
\quad
\Rightarrow
\quad \varphi_\beta\circ\varphi_\alpha^{-1}:\varphi_\alpha(U_\alpha\cap U_\beta)\to \varphi_\beta(U_\alpha\cap U_\beta)$ is smooth.(iv) Countably many of the sets $U_\alpha$ cover $M$.
(v)
$
\left.
\begin{array}{c}
p,q\in M\\
p\neq q
\end{array}
\right\}
\quad
\Rightarrow
\quad
\left\{
\begin{array}{c}
\exists \alpha\in J\text{ such that } p,q\in U_\alpha,\quad\text{ or}\\
\exists \alpha,\beta\in J\text{ such that } p\in U_\alpha, q\in U_\beta \text{ and } U_\alpha\cap U_\beta=\emptyset
\end{array}
\right.
$Then $M$ has a unique manifold structure such that each pair $(U_\alpha,\varphi_\alpha)$ is a smooth chart.
PROOF. Let $\mathcal B=\{\varphi_\alpha^{-1}(V):\alpha\in J, V\text{ open in } \mathbf R^n\}$.
Claim 1: $\mathcal B$ forms a basis for $M$.
Proof: We use $(i)$—$(iv)$ in this proof.
From $(iv)$ we see that the elements of $\mathcal B$ cover $M$.
Now let $\varphi_\alpha^{-1}(V)$ and $\varphi_\beta^{-1}(W)$ be two elements of $\mathcal B$, where $V$ and $W$ are open in $\mathbf R^n$.
To show that $\mathcal B$ forms a basis, it is enough to show that $ \varphi_\alpha^{-1}(V)\cap\varphi_\beta^{-1}(W)$ itself lies in $\mathcal B$.
Note that
\begin{equation*}
\varphi_\alpha^{-1}(V)\cap \varphi_\beta^{-1}(W)=\varphi_\alpha^{-1}\Big(V\cap(\varphi_\beta\circ\varphi_\alpha^{-1})^{-1}(W)\Big)
\tag{1}
\end{equation*}
But by (iii), $\varphi_\beta\circ\varphi_\alpha^{-1}$ is continuous, and therefore $(\varphi_\beta\circ\varphi_\alpha^{-1})^{-1}(W)$ is open in $\varphi_\alpha(U_\alpha\cap U_\beta)$.
By (ii), $\varphi_\alpha(U_\alpha\cap U_\beta)$ is open in $\mathbf R^n$ and therefore $(\varphi_\beta\circ\varphi_\alpha^{-1})^{-1}(W)$ is open in $\mathbf R^n$.
Using this in $(1)$, we immediately see that $\varphi_\alpha^{-1}(V)\cap\varphi_\beta^{-1}(W)$ is in $\mathcal B$.
This settles the claim.
Let $\tau$ be the topology generated on $M$ by $\mathcal B$.
By definition of $\mathcal B$, each function $\varphi_\alpha$ is a homeomorphism onto its image.
Thus $(M,\tau)$ is locally Euclidean of dimension $n$.
Claim 2: $(M,\tau)$ is Hausdorff.
Proof: This uses $(v)$. Let $p,q\in M$ with $p\neq q$. If $\exists \alpha,\beta\in J$ such that $p\in U_\alpha, q\in U_\beta$ and $U_\alpha\cap U_\beta=\emptyset$, then we have nothing to prove. The other possibility if that $\exists \alpha\in J$ such that $p,q\in U_\alpha$.
Now since $\varphi_\alpha(U_\alpha)$ is open in $\mathbf R^n$, there exist disjoint open sets $V$ and $W$ open in $\varphi_\alpha(U_\alpha)$ containing $p$ and $q$ respectively. The neighborhoods $\varphi_\alpha^{-1}(V)$ and $\varphi_\alpha^{-1}(W)$ separate $p$ and $q$ in $M$. Thus the claim is settled.
Claim 3: $(M,\tau)$ is second countable.
Proof: Note that since $\varphi_\alpha(U_\alpha)$ is second countable, and since $\varphi_\alpha:U_\alpha\to\varphi_\alpha(U_\alpha)$ is a homeomorphism, we must have $U_\alpha$ is second countable. The proof is now immediate from $(iv)$ and Lemma given at the bottom. The above working shows that $(M,\tau)$ is a topological $n$-manifold. Now from $(iii)$ it is clear that $\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in J}$ is a smooth atlas on $M$, giving $M$ a smooth structure.
Now we need to establish that this is the only smooth structure on $M$ such that each $\varphi_\alpha:U_\alpha\to\varphi_\alpha(U_\alpha)$ is a smooth chart on $M$ and here I am stuck.
In fact what Lee writes is that "It is clear that this topology and smooth structure are the unique ones satisfying the conclusions (conditions?) of the lemma."
Can somebody please explain this to me.
LEMMA. Let $X$ be a topological space and $\{U_n\}_{n\in\mathbf N}$ be a countable open cover of $X$ such that each $U_i$ is second countable in the subspace topology. Then $X$ is second countable.
Best Answer
About the topology: Every $(U_\alpha,\varphi_\alpha)$ needs to be a chart, meaning that $\varphi_\alpha$ is a homeomorphism on its image, so for every subset $V\subset U_\alpha$, $V$ is open if and only if $\varphi_\alpha(V)\subset\mathbb{R}^n$ is open. Since the $U_\alpha$s cover $M$, it means that the collection of open subsets of elements in $\{\varphi_\alpha(U_\alpha)\}$ induces a basis of a topology on $M$, thus the topology is indeed unique.
Regarding the smooth structure: The thing is that given an atlas $\{U_\alpha,\varphi_\alpha\}$, it always determines a unique smooth structure. To see this, note that if $(V,\psi),(V',\psi')$ both "agree" with the given atlas (i.e. all obtained transition maps are smooth), it follows from the chain rule that these two charts also agree with each other. Hence there is no choice when extending an atlas to a smooth structure - one just adds every chart that agrees with the given atlas.
Once again, since the $U_\alpha$'s cover $M$, the given collection is an atlas and it thus induces a unique smooth structure.