How can there be isolated points in an open set (in the definition of meromorphic)?

Question

The definition of a function $f: O \to \Bbb{C}$ from a an open subset $O$ of $\Bbb{C}$ is meromorphic if it is holomorphic (complex differentiable) on all of $O$ except for a set of isolated points. How can you have isolated points here? Since any neighborhood around the points that you take will have a point of $O$ in it.

Answer 1

The neighbourhood will have other points of $O$ in it, but not other points of the set. For example, $f(z) = 1/z + 1/(z-1)$ is meromorphic in $O = \mathbb C$. The set of isolated points is $\{0,1\}$ in this case.