Simply connected domain: Difference between revisions
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* Every holomorphic function is integrable, i.e. has a global primitive | * 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 | ||
The [[Riemann mapping theorem]] states that any simply connected domain that is not the whole of <math>\mathbb{C}</math> is [[biholomorphically equivalent domains|biholomorphically equivalent]] to the [[open unit disk]]. | |||
Revision as of 19:12, 26 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
The Riemann mapping theorem states that any simply connected domain that is not the whole of is biholomorphically equivalent to the open unit disk.