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Suppose $N$ is a manifold of dimension $n$. Now a regular submanifold $S$ of $N$ of dimension $k$ is defined as, if for every point $p$ of $S$ there is a coordinate chart $(U,u_*)$ from a maximal atlas such that $U \cap S$ is defined as the vanishing of $n-k$ coordinate functions on $U$.

My question is whether the dimension of the submanifold is uniquely defined or not? If yes why is it so?

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Usually the dimension being constant is axiomatized in all those definitions of manifolds/submanifolds, sometimes not explicitly. In any case, it's easy to see that, given your definition, any point of $S$ has a neighborhood homeomorphic to $\mathbb{R}^k$, for some (possibly varying) $k$. If, in addition, $S$ is connected, it follows from the Invariance of Domain theorem that $k$ never changes. If $S$ is actually a differentiable manifold (say, because $M$ is), then this follows much more easily from the fact that open subsets of euclidean spaces of different dimensions can't be diffeomorphic.

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