The usual topology for the complex plane $\sC$ is the topology induced by the metric $$d(x,\,y) := |x\!-\!y|$$ for $x,\,y \in \sC$ . Here, $|\cdot|$ is the complex modulus.

If we identify $\sR^2$ and $\sC$ , it is clear that the above topology coincides with topology induced by the Euclidean metric on $\sR^2$ .

Some basic topological concepts for $\sC$ :

1. The open balls $$B_r(\zeta) \;=\; \{z\in\sC\,\vdots\; |z\!-\!\zeta| < r\}$$ are often called open disks.
2. A point $\zeta$ is an accumulation point of a subset $A$ of $\sC$ , if any open disk $B_r(\zeta)$ contains at least one point of $A$ distinct from $\zeta$ .
3. A point $\zeta$ is an interior point of the set $A$ , if there exists an open disk $B_r(\zeta)$ which is contained in $A$ .
4. A set $A$ is open, if each of its points is an interior point of $A$ .
5. A set $A$ is closed, if all its accumulation points belong to $A$ .
6. A set $A$ is bounded, if there is an open disk $B_r(\zeta)$ containing $A$ .
7. A set $A$ is compact, if it is closed and bounded.