Assume M is a manifold, S is a subset of M. If to any point x of S, there is an open set U in M s.t. U$\cap$S is an sub manifold of U, then S is a submanifold of M?
I arose this question from: To prove the regular value theorem, It is enough to show that for every point a of $f^{-1}$(c) there is a neighborhood U of a such that U ∩ $f^{−1}$(c) is a submanifold of U of dimension m − n (where c is the regular point).