There are some questions about Lang's definition of differentiable manifold, but I think they does not ask the same (I think everybody think that).
My definition is taken from Fundamentals of differential geometry, Springer (1999). The definition is the following:
Let $X$ be a set. An atlas of class $C^p$ ($p\geq0$) on $X$ is a collection of pairs $(U_i,\varphi_i)$ ($i$ ranging in some index set), satisfying the following conditions:
AT 1. Each $U_i$ is a subset of $X$ and the $U_i$ cover $X$.
AT 2. Each $\varphi_i$ is a bijection of $U_i$ onto an open subset $\varphi_iU_i$ of some Banach space $E_i$ and for any $i$, $j$, $\varphi(U_i\cap U_j)$ is open in $E_i$.
AT 3. The map $$ \varphi_j\varphi_i^{-1}:\varphi_i(U_i\cap U_j) \rightarrow \varphi_j(U_i\cap U_j) $$ is a $C^p$-isomorphism for each par of indices $i$, $j$.
This definitions is clear. The problem cames in the next claim:
It is a trivial exercise in point set topology to prove that one can give $X$ a topology in a unique way such that each $U_i$ is open, and the $\varphi_i$ are topological isomorphisms.
The unique trivial way I know to carry such a topology on $X$ is defining $\tau$ as the collection of all $\varphi_i^{-1}A_i$ for every open subset $A_i\subset\varphi_iU_i$ and for every index $i$.
If I would be right, how could I show this topology is unique?
Thanks
That seems correct. Basically, this is the initial topology.
Showing that it is well defined is basically applying the fact that the image of each open set must also be open for these guys to be topological isomorphisms.