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.
Thanks in advance.