Show, that the intersection of two manifolds $M, N \subset \mathbb{R}^n$ doesn't need to be a manifold [closed]

differential-geometry manifolds

I've got some ideas on how to tackle it, but I don't know what I should do:

  1. To choose such a $M$ and $N$ that it's intersection is the empty set (or a set of meassure zero maybe, maybe something with the Cantor set, or to choose $M$ and $N$ as a countable set, or choose $M$ or $N$ as just the empty set?)

  2. To choose such a $M$ and $N$, that there exists some $x \in M \cap N$ that it's transition function is not a homomorphism (might be trickier to show that)

I don't know how to start here, but here are my ideas. Any suggestions?


Hint

Look at the intersection of the lateral surfaces of two cylinders with the same radii, coplanar and orthogonal axis.

The intersection is not a manifold. It is however the union of two manifolds.

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