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This question may be repetitive and I have read some questions on the definition of submanifolds, but I was not satisfied with the answers.

There are some types of submanifolds---immersed, embedded, and regular, as far as I know, and the first includes the other two as special cases. The definition of immersed submanifolds is rather complicated and hard to digest---at least for me.

What would go wrong if we defined a submanifold as follows?

Suppose that $M$ is a smooth manifold with coordinate neighborhoods $U_{\alpha}, \varphi_{\alpha}$ for every point of $M$. Then a subset $N$ of $M$ with the subspace topology is called a submanifold with coordinate neighborhoods $N \cap U$ and $\varphi |_{N \cap U}$.

Would anyone please help me out? Thanks.

Behrooz
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    Any subset? If you consider $N={(x,y)\in\mathbb{R}^2:xy=0}$ according to your definition it would be a "smooth" submanifold. – mfl Dec 23 '16 at 15:02
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    Your definition fails to capture the key idea: given an $m$-dimensional manifold $M$ and given $n<m$, an $n$ dimensional submanifold $N \subset M$ looks, locally, like $\mathbb{R}^n$ embedded as a coordinate subspace of $\mathbb{R}^m$. – Lee Mosher Dec 23 '16 at 19:14

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