topology of the complex plane
The usual topology for the complex plane
is the topology
induced by the metric
for  .
Here, is the complex modulus (http://planetmath.org/ModulusOfComplexNumber).
If we identify and , it is clear that the above topology coincides with topology induced by the Euclidean metric on .
Some basic topological concepts for :
- 1.
-
2.
A point is an accumulation point of a subset of , if any open disk contains at least one point of distinct from .
-
3.
A point is an interior point of the set , if there exists an open disk which is contained in .
-
4.
A set is open, if each of its points is an interior point of .
-
5.
A set is closed, if all its accumulation points belong to .
-
6.
A set is bounded
, if there is an open disk containing .
-
7.
A set 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 |