Carathéodory completeness on the plane

Abstract

. M. A. Selby [ 8–10 ] and, independently, N. Sibony [ 11 ] proved that on the complex plane c -completeness is equivalent to c -finitely com-pactness. Their proofs are quite similar and are based on [ 4 ]. We give more refined equivalent conditions and, along the way, simplify the proofs.