topology of the complex plane

The usual topology for the complex plane $ℂ$ is the topology induced by the metric

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

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

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

Some basic topological concepts for $ℂ$:

1. The open balls

   are often called open disks.

2. A point $\zeta$ is an accumulation point of a subset $A$ of $ℂ$, 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	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