By differentiable structure, I mean a collection of injections of n-th Euclidean space to some set $M$, where the domains are open in $\mathbb{R}^n$, the images cover $M$, and mappings from non-empty intersections of the images are smooth.
I am adding a twist, that the covering is actually a partition, so the second condition of a differentiable structure is true in a vacuum. I am not questioning that this is in fact a smooth manifold, since it is.
My question is, what can we say about such a manifold? I thought that maybe it is always diffeomorphic (there exists a smooth mapping with smooth inverse) to some subspace of $\mathbb{R}^n$, more concretely, the space we get by union of all domains of the injections. So I tried to construct such a diffeomorphism, but got stuck trying to prove that the domains are also a partition of some subset of $\mathbb{R}^n$, and after the fact I see no reason as to why they should be. Without domains being a partition, defining a diffeomorphism becomes difficult. Maybe there is Choice involved? Not sure.
Also, what other properties can be derived from the hypothesis of the covering being a partition?
It seems more of a set-theoretic question than a topology question, unless the fact that the domains being open plays a role, which as far as I can see, it does not.
As an example, take $M = \{ f_t \}_{t \neq 0}$ where $f_t (x) = tx$, and cover it by $F_+(t) = f_t, t>0$ and $F_-(t) = f_t, t<0$. I split up $\mathbb{R}^* = \mathbb{R}-\{0\}$ on purpose, to make it even clearer that change of parameters holds in a vacuum. As is, $M$ is diffeomorphic to $\mathbb{R}^*$.