Let $U\_1, U\_2$ be open subsets of $\mathbb{R}^n$. Both are naturally differentiable submanifold, getting the differentiable structure from $\mathbb{R}^n$. Further, both are natural topological manifolds, as submanifolds of $\mathbb{R}^n$.


**Question:**
If $U\_1$ and $U\_2$ are **homeomorphic**, are they also **diffeomorphic**?


Of course two general topological manifolds which are homoemorphic do not need to be diffeomorphic. But here the differentiable structure is a very special one.


The answer might depend on the dimension $n$. For $n=1,2,3$ it is yes, as there each topological manifold has a unique differentiable structure. For $n \geq 5$ and $U\_1$ an open ball the answer is yes by the uniqueness of differentiable structures on $\mathbb{R}^n$ for $n \geq 5$.


Some special cases are:


1. What happens if $U\_1$ (and hence $U\_2$) is contractible?
2. What happens if $U\_1$ is a ball and $n=4$? Is there an exotic $\mathbb{R}^4$ which can be realized as an open subset of the standard $\mathbb{R}^4$?


(The question came up because I encountered a sloppy definition of a manifold. One can view the above manifolds as being defined by only one chart. [That of course depends on your definition of chart, if you require it to start from a ball or not.] So the questions basically asks: How do manifolds with only one chart look like?)



Are homeomorphic open subsets of $\mathbb{R}^n$ also diffeomorphic?
In fact, there exist uncountably many *small* exotic smooth $\mathbb{R}^4$'s, i.e. smooth manifolds $X$ which are homeomorphic to $\mathbb{R}^4$ but not not diffeomorphic to it and which can be smoothly embedded as open subsets of $\mathbb{R}^4$. There are discussions of this in many places; I recommend first reading the appropriate part of Scorpan's book "The Wild World of 4-Manifolds" for a brief survey (it includes a nice bibliography of more detailed sources). 


