Classical and quantum surgery of geometries in an open inflationary universe

We study classically and quantum mechanically the Euclidean geometries comp atible with an open inflationary universe of a Lorentzian geometry. The Lor entzian geometry of the open universe with all ordinary matter state matche s either an open or a closed Euclidean geometry at the cosmological singula rity. The open Lorentzian geometry with ail exotic matter state matches onl y the open Euclidean geometry and describes a genuine instanton regular at a boundary of finite radius. The wave functions that describe the quantum c reation of the open inflationary universe are found.