The natural topology on a $C^k$-manifold Lang- Differential Manifolds p.21

First Question. As it is written above, the author says it is trivial to prove that one can give $X$ a topology in a unique way such that $U_i$ is open, and the $\phi_i$ are topological embeddings, but how?
Second Question. Let $\mathscr{A},\mathscr{B}$ be $C^k$-atlases on $X$ and let $T_1,T_2$ be the natural topologies on $X$ (as in the first question) induced by $\mathscr{A},\mathscr{B}$. If these two atlases are equivalent, then is $T_1=T_2$?
Third Question. Let $\mathscr{A}$ be a $C^k$-atlas on $X$ and let $T$ be the natural topology on $X$ induced by this atlas. Let $\mathscr{T}$ be the topology generated by the domains of charts of all the $C^k$-atlases equivalent to $\mathscr{A}$. Then, is $\mathscr{T}=T$?
Thank you in advance!

The topology you want is defined as follows: $V \subset M$ is open iff $\varphi_i(V \cap U_i)$ is open in $\mathbf E_i$ for all $i$.
With this definition, the answers to the next two questions are clearly "yes".
My favourite textbook on this sort of stuff is Berger and Gostiaux - see 2.2.6, 2.2.7 and 2.2.8, which provide proofs of your three claims.
