Topology of the Complex Plane

Theorem: Topology of the Complex Plane

The function $d:\mathbb{C}\times \mathbb{C}\to \mathbb{R}$ defined as

$$d(z_1,z_2)=|z_2-z_1|$$

is a metric on the complex numbers.

PROOF

TODO

NOTE

As a metric, $d$ induces a topology on $\mathbb{C}$ and, unless otherwise specified, all topology-related terminology relates to this induced topology.

Neighborhoods

Definition: $r$-Neighborhood

Let $r>0$ and $z\in \mathbb{R}$.

The $r$-neighborhood of $z$ is the open ball $B_r(z)$, i.e.

$$B_r(z)=\{x\in \mathbb{C}\mid |x-z|<r\}$$

Definition: Deleted $r$-Neighborhood

Let $r>0$ and $z\in \mathbb{R}$.

The deleted $r$-neighborhood of $z$ is its $r$-neighborhood without $z$ itself:

$$B_r(z)\setminus \{z\}=\{x\in \mathbb{C}\mid 0<|x-z|<r\}$$

Properties

Theorem: Boundary and Interior of Neighborhoods

Let $r>0$ and $z\in \mathbb{C}$.

The boundary of the $r$-neighborhood and the deleted $r$-neighborhood of $z$ is the disk

$$\{x\in \mathbb{C}\mid |x-z|=r\}$$

PROOF

TODO