topology of the complex plane

The usual topology for the complex planeMathworldPlanetmath is the topologyMathworldPlanetmath induced by the metric


for  x,y. Here, || is the complex modulusMathworldPlanetmath (

If we identify 2 and , it is clear that the above topology coincides with topology induced by the Euclidean metric on 2.

Some basic topological concepts for :

  1. 1.

    The open ballsPlanetmathPlanetmath


    are often called open disks.

  2. 2.

    A point ζ is an accumulation point of a subset A of , if any open disk Br(ζ) contains at least one point of A distinct from ζ.

  3. 3.

    A point ζ is an interior point of the set A, if there exists an open disk Br(ζ) 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 boundedPlanetmathPlanetmathPlanetmath, if there is an open disk Br(ζ) 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