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- Mar 10, 2012

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**Question:**Let $M$ be a $k$-manifold-without-boundary in $\mathbb R^n$ and $N$ be another manifold-without-boundary in $\mathbb R^n$ such that $M\subseteq N$.

Assume that there exists a point $\mathbf p\in M$ such that each neighborhood $U$ of $\mathbf p$ has a point $\mathbf q\in U$ such that $\mathbf q\in N\setminus M$.

Then can $N$ possibly be a $k$-manifold?

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Intuitively it seems obvious that the dimension of $N$ should be greater than $k$ but I haven't been able to make any progress to make it into a proof.

Can somebody help?

Thanks.