# Is the union of close disjoint set connected or not?

For a set to be connected, it should have the property of not being able to be presented as a union of open disjoint sets(with respect to some topology- that I didn't quite get). I want to show that a ring $S=\{r<|z-z_0|<R\}$ is not simply connected by showing $\Bbb{C}\setminus S$ is not connected, but according to my notion of open sets, this set cannot be the union of two open sets as it is a closed set, but intuitively I have no doubt the is a "disconnection". Any clarification of the issue will help me.

A topological space $X$ is called connected if the only sets which are both open and closed are $X$ and $\emptyset$. Otherwise, if $V$ is open and closed, then so is $X\setminus V$, so by your definition, $X$ can be expressed as a union of disjoint open sets.
In your definition, you are considering your set to be a subset of a topological space endowed with the subspace topology. So your open sets are those which are the intersection of an open set on the whole topology (i.e. $\mathbb C$) with your set.