Invariance of the number of holes

Question

Suppose that $\Omega$ is an open subset of $\mathbb{R}^2$. What conditions on $\Omega$ ensure that $\Omega_k$ homeomorphic to $\Omega_l$ implies $k=l$, where $\Omega_k$ denotes $\Omega$ with any $k$ points removed. I know it to be true for $\Omega=\mathbb{R}^2$ (consider the first betti number), but what about the more general situation? In the end this will give a good reason why one should think of the first Betti number as being the number of holes of thst space.

Answer 1

One thing you'll want to assume is that $\Omega$ is connected. Two disjoint disks, each with one point removed, is not homeomorphic to two disks where one has two points removed.