Suppose $M$ is a smooth complete Riemannian manifold and $x$ is a point in $M$. For any positive radius $r$ we consider the open ball $B(x,r)$ centered at $x$ with radius $r$.

If we ignore the Riemannian structure on $B(x,r)$ and consider it only as a smooth manifold. Are there only a countable number of diffeomorphism classes it can belong to?

An afirmative answer includes as a particular case the fact that there are only countably many compact manifolds up to diffeomorphism. This follows from Cheeger's finiteness theorem because on any such manifold has bounded curvature, volume, and injectivity radius.

It is well known that there are uncountably many diffeomorphism classes of smooth manifolds (even homeomorphic to $\mathbb{R}^4$; Are there uncountably many surfaces?).

My motivation for this is that I'm trying to study a particular concept of a random complete Riemannian manifold with a distinguished point. The starting point is to topologize the space of such manifolds (and I need something stronger then Lipschitz) and things would be simpler if the answer to this question was yes (and I had a short proof or reference :) ). Also, the question seems interesting in its own right.