# If you remove a submanifold of codimension 1 from its manifold do you get a manifold?

Suppose I have a manifold $\mathcal{M}$ and a closed submanifold $\mathcal{N} \subset \mathcal{M}$ of codimension 1. If I remove the closed submanifold $\mathcal{N}$ from $\mathcal{M}$ will I be left with a manifold?

I am not sure if it is true but it looks very plausible. However, I am pretty sure that if would only hold for codimension 1. For example in the manifold $\mathbb{R}^2$ I can take the submanifold complement to a figure "8".

Furthermore, would the statement also be true for smooth manifolds or symplectic manifolds?

Any help is welcome.

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No, for example remove $(0,1) \times \{0\}$ from $\mathbb{R}^2$. –  Chris Eagle Jun 3 '12 at 16:58
@ChrisEagle What about if you remove a closed submanifold? –  JSchlather Jun 3 '12 at 16:59
Then the result is open, hence clearly a manifold. –  Chris Eagle Jun 3 '12 at 17:00
But open subsets of manifolds are always manifolds, right? Just restrict the charts. Also, it's not obvious to me that the figure 8 is a manifold. –  Dylan Moreland Jun 3 '12 at 17:01
@user29751: But it's not just that you can remove a closed manifold, you can even remove any closed set. So, if something like a Cantor set is removed, the resulting thing is still a manifold, being an open subset of a manifold. –  Jason DeVito Jun 3 '12 at 17:32