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

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.

Under what conditions will the rectangle of the Japanese theorem be a square?

Is $50$th percentile equal to median?

show that $a_{n+874}=a_{n}$,if such $a_{n+2}=\left\lceil \frac{4}{3}a_{n+1}-a_{n}+0.5\right\rceil$

$n=pq$ with $p<q$ odd primes is there a fast way to determine if $p-1|q-1$ without knowing $p$ and $q$

Factoring $1+x+\dots +x^n$ into a product of polynomials with positive coefficients

Do there exist four polynomials with real coefficients so that ...

A simple game on infinite chessboard

The best $n$-digit password?

Evaluate $\int_{(-\infty,\infty)^n}\frac{\prod_{k=1}^n \sin(a_k x_k)}{\prod_{k=1}^n x_k}\frac{\sin(\sum_{k=1}^n a_k x_k)}{\sum_{k=1}^n a_k x_k}$

$|f(f(z))-z^2|$ must be large somewhere in the disc $\mathbb{D}$?

Proving finite dimensionality of modular forms using representation theory?