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
• The complement of the domain in the Riemann sphere is a connected set