Hi! Probably this is an easy question, but i can't see the answer.
Let $X$ be a a smooth real manifold with $\dim(X)=d$ and $M,N\subset X$ two smooth submanifolds with $\dim(M)=m$ and $\dim(N)=n$. The submanifolds $M,N$ intersect but not transversely.
What can i say about connected components of $M\cap N$? More precisely, is it possible to find three manifolds $X,M,N$ as above such that a connected component of $M\cap N$ is not a manifold? Or a connected component that is not smooth? (In all the examples i thought, connected components of $M\cap N$ were smooth)
Thank you in advance.