Simply connected domain

Definition

A domain (open connected subset) in $\mathbb {C}$ 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 $\mathbb {C}$ is biholomorphically equivalent to the open unit disk.