Topology of the complex plane

Since \(|z-w|\) is the distance between two complex numbers, \(\mathbb{C}\) behaves like the plane \(\mathbb{R}^2\). A set \(U\) is open if each of its points is the center of some disk \(D(z_0,r)\) contained in \(U\); it is closed if its complement is open, or equivalently, if it contains all of its boundary points. A disk \(|z-z_0|<r\) is open; with \(\le\) it is closed.

The boundary of a set consists of the points with both points of the set and points outside arbitrarily close: that of the disk \(|z|<1\) is the circle \(|z|=1\). A set is bounded if it fits in a disk \(|z|<R\) for some \(R\). Open and closed are not opposites: \(\mathbb{C}\) and the empty set are both, and a set like \(\{|z|<1\}\cup\{1\}\) is neither.

An open set is connected if any two of its points can be joined by a polygonal path inside it. A connected open set is called a domain (or region), and it is the usual setting of the theory: the annulus \(1<|z|<2\) and the half-plane \(\operatorname{Re}z>0\) are domains; \(\operatorname{Re}z\neq0\) is open but not connected.

A typical mistake is to think a closed set is «one that is not open». Another is forgetting that the sign \(<\) or \(\le\) decides whether the circle belongs. This notion is used right away to say where a complex function is defined and what it means to approach a point.