Suppose that $M$ is a smooth manifold of dimension $m$, and $N$ is a topological subspace.
Definition 1) $N$ is called a smooth submanifold of $M$ provided that for each $x\in N$ there exists a chart $(U,\psi)$ in the maximal atlas for $M$ with $x\in U$ such that $\psi(U\cap N)=\psi(U) \cap (\mathbb R^n \times {0_{\mathbb R^{m-n}}})$ for some $n\le m$.
Definition 2) $N$ is called a smooth submanifold of $M$ provided that for each $x\in N$ there exists a chart $(V,\phi)$ in the maximal atlas for $M$ with $x\in V$ such that $V\cap N= \phi^{-1}(\mathbb R^n \times {0_{\mathbb R^{m-n}}})$ for some $n\le m$.
My question is that if these two definitions are equivalent or not. If they are equivalent, I will try to prove it myself. If they are not equivalent, I would need a counterexample.