https://math.stackexchange.com/a/297673/519033
"More generally: there is a holomorphic bijection from the open unit disk onto a region U if and only if U is simply connected and the complement of U (in the Riemann sphere) has at least two points."
"More generally: there is a holomorphic bijection from the open unit disk onto a region U if and only if U is simply connected and the complement of U (in the Riemann sphere) has at least two points."