Simply connected domain

A domain is simply connected when every closed path in it can be continuously contracted to a point while remaining in the domain.

A crosscut of a planar domain is a simple open arc inside the domain whose two endpoints are distinct boundary points, interpreted as prime ends when necessary. In a simply connected domain it divides the domain into two components. In the half-plane, a bounded crosscut joining two real points separates the intervening real interval from infinity; a Loewner chain completing that crosscut swallows this interval.

Every holomorphic function $f$ on a simply connected domain has a single-valued antiderivative. Fixing $z_0$ and settinggives a path-independent function with $F^{\prime }=f$, because the Cauchy integral theorem makes the integral around every closed path zero.