topology of the complex plane
The usual topology for the complex plane^{} $\u2102$ is the topology^{} induced by the metric
$$d(x,y):=xy$$ 
for $x,y\in \u2102$. Here, $\cdot $ is the complex modulus^{} (http://planetmath.org/ModulusOfComplexNumber).
If we identify ${\mathbb{R}}^{2}$ and $\u2102$, it is clear that the above topology coincides with topology induced by the Euclidean metric on ${\mathbb{R}}^{2}$.
Some basic topological concepts for $\u2102$:
 1.

2.
A point $\zeta $ is an accumulation point of a subset $A$ of $\u2102$, 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.
Title  topology of the complex plane 
Canonical name  TopologyOfTheComplexPlane 
Date of creation  20130322 13:38:40 
Last modified on  20130322 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 3000 
Related topic  IdentityTheorem 
Related topic  PlacesOfHolomorphicFunction 
Defines  open disk 
Defines  accumulation point 
Defines  interior point 
Defines  open 
Defines  closed 
Defines  bounded 
Defines  compact 