Let $\Phi=\{\phi_i , U_i\}_{i\in \Lambda}$ be an atlas on an $n$-dimensional manifold $M$. Put $\phi_i(U_i)=V_i\subset \mathbb{R}^n$ and let $X$ be the identification space obtained from $\bigcup_{i\in \Lambda}V_i\times i$ (disjoint union) when $(x,i)\sim (\phi_j\phi_i^{-1}(x),j)$. Show that $X$ is homeomorphic to $M$.
I would like some guidance for the proof. Should I induce quotient topology on $X$? Is it also true that $X$ is diffeomorphic to $M$? How does one induce a differentiable structure on $X$?