Submanifold separates the manifold

Question

I believe that if we delete the small neighbourhood of a submanifold with codimension more than 2 from the ambient manifold, it won’t separate the manifold. Is there any topological argument I can use? Like proving $H_0=0$ ? Maybe this is not true in general, a counter example will be good.

Answer 1

Here is a topological argument you can use: if $N\subset M$ is a codimension $\geq 2$ submanifold, then it does not locally separate $M$. So you just need to show that $\mathbb{R}^{m}\subset\mathbb{R}^n$, for $m\leq n-2$, doesn't separate.