# On limits of manifolds

This question should be fairly elementary. I’d just like to check I’m not missing anything.

Let $$\{M_n\}_{n\ge 0}$$ be an inverse system of smooth manifolds with transition maps $$f_{t,s} : M_t\to M_s$$, $$t\ge s$$, that are local diffeomorphisms.

Let $$M$$ be the topological inverse limit.

For every $$x\in M$$, does there exist an open neighborhood $$U\subset M$$ of $$x$$, such that $$U$$ is homeomorphic to some smooth manifold that is diffeomorphic to an open subset of $$M_n$$ for some $$n$$?

Many of these are obtained by taking the inverse limit of finite-sheeted covers of a fixed manifold. The universal such example is obtained by taking a manifold $$M$$ with infinite residually finite fundamental group $$\pi_1(M)$$, and taking its profinite completion $$\widehat{\pi_1(M)}$$. Let $$\tilde{M}$$ denote the universal cover of $$M$$. Then $$(\tilde{M}\times \widehat{\pi_1(M)})/ \pi_1(M)$$ is a solenoid, where the action is by covering translation on the left and coset action on the right where $$\pi_1(M)\subset \widehat{\pi_1(M)}$$.
This space is locally homeomorphic to open subsets of $$\mathbb{R}^n \times \widehat{\pi_1(M)}$$. For an infinite residually finite group, $$\widehat{\pi_1(M)} \cong \mathcal{C}$$, the Cantor set. So there is a neighborhood basis of sets which are homeomorphic $$\mathbb{R}^n \times \mathcal{C}$$ where $$\mathcal{C}$$ is totally disconnected, since $$\mathcal{C}$$ has a neighborhood basis of sets homeomorphic to itself.