# topology of the complex plane

The usual topology for the complex plane $\mathbb{C}$ is the topology induced by the metric

 $d(x,\,y):=|x\!-\!y|$

for  $x,\,y\in\mathbb{C}$. Here, $|\cdot|$ is the complex modulus (http://planetmath.org/ModulusOfComplexNumber).

If we identify $\mathbb{R}^{2}$ and $\mathbb{C}$, it is clear that the above topology coincides with topology induced by the Euclidean metric on $\mathbb{R}^{2}$.

Some basic topological concepts for $\mathbb{C}$:

1. 1.

The open balls

 $B_{r}(\zeta)\;=\;\{z\in\mathbb{C}\,\vdots\;|z\!-\!\zeta|

are often called open disks.

2. 2.

A point $\zeta$ is an accumulation point of a subset $A$ of $\mathbb{C}$, if any open disk $B_{r}(\zeta)$ contains at least one point of $A$ distinct from $\zeta$.

3. 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. 4.

A set $A$ is open, if each of its points is an interior point of $A$.

5. 5.

A set $A$ is closed, if all its accumulation points belong to $A$.

6. 6.

A set $A$ is bounded, if there is an open disk $B_{r}(\zeta)$ containing $A$.

7. 7.

A set $A$ is compact, if it is closed and bounded.

 Title topology of the complex plane Canonical name TopologyOfTheComplexPlane Date of creation 2013-03-22 13:38:40 Last modified on 2013-03-22 13:38:40 Owner matte (1858) Last modified by matte (1858) Numerical id 8 Author matte (1858) Entry type Definition Classification msc 54E35 Classification msc 30-00 Related topic IdentityTheorem Related topic PlacesOfHolomorphicFunction Defines open disk Defines accumulation point Defines interior point Defines open Defines closed Defines bounded Defines compact