Let $X,Y \subset B(0,1)$ be two subsets of the unit open ball $B(0,1) \subset \mathbb R^n$, each of them diffeomorphic to $B(0,1)$. Is there any classification for the intersection $X\cap Y$?
More precisely, is it true that $X\cap Y$ has a finite number (or at least countable number) of connected components, each of which is diffeomorphic to $B(0,1)$ in $\mathbb R^n$ ?
Just for an example, in dimension $2$, each connected component of $X\cap Y$ would be an open simply connected subset, thus by the Riemann mapping theorem (see https://en.wikipedia.org/wiki/Riemann_mapping_theorem) it will be diffeomorphic to the open unit ball. Although this result is only in the two-dimensional setting, the starting hypothesis is much weaker then the ones I posed above.
Therefore, my question is in some sense if there is a higher dimensional weak generalisation of such result.
[Edit.] Gae. S. gave an example that showed that the answer to 1 is not true.
Take a hollow rubber ball in ${\mathbb R}^3$, let the air go out, and press one half of the surface into the hollow of the other half. Including the interior you now have a body which is homeomorphic to $B(0,1)\subset{\mathbb R}^3$, but looks like a cup.
Make a second copy of such a thing.
Now intersect the two rim zones of your "cups". The intersection is a full donut, and is not homeomorphic to $B(0,1)$.