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:

What happens if $U_1$ (and hence $U_2$) is contractible?

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?)