Simply connected domain: Difference between revisions
(New page: ==Definition== A domain (open connected subset) in <math>\mathbb{C}</math> is termed a '''simply connected domain''' if it satisfies the following equivalent conditions: * It is simp...) |
No edit summary |
||
| Line 6: | Line 6: | ||
* Its first homology group is trivial | * Its first homology group is trivial | ||
* Any cycle in it is zero-homologous: it does not wind around any point in the complement of the domain | * Any cycle in it is zero-homologous: it does not wind around any point in the complement of the domain | ||
* Every holomorphic function is integrable, i.e. has a global primitive | |||
* The complement of the domain in the [[Riemann sphere]] is a connected set | * The complement of the domain in the [[Riemann sphere]] is a connected set | ||
Revision as of 20:02, 20 April 2008
Definition
A domain (open connected subset) in is termed a simply connected domain if it satisfies the following equivalent conditions:
- It is simply connected as a topological space i.e. its fundamental group is trivial
- Its first homology group is trivial
- Any cycle in it is zero-homologous: it does not wind around any point in the complement of the domain
- Every holomorphic function is integrable, i.e. has a global primitive
- The complement of the domain in the Riemann sphere is a connected set