1) U is an open connected set. Open meaning it contains none of its boundary points, and connected meaning that it cannot be represented as two disjoint nonempty open subsets. In other words, if you partitioned the set (say in half), each point on the boundary of the subsets would be in either one set or the other.
U is the set of all complex numbers in this arbitrary set, that satisfies the above properties.
2/3) Theorem 1.1 seems to indicate that it is a pretty basic Theorem?