Biholomorphic approximation of planar domains

Abstract

This paper establishes the existence of a domain (open connected subset) B of the complex plane C such that for every domain c C and every compact set K c 9 there is a biholomorphic embedding e: B - , such that K c e(B) c cl [e(B)] c . 1* Introduction* Let x and 2 be domains (i.e., open connected sets) in the complex plane C such that cl , c 2 (cl = closure). A domain is a biholomorphic approximation of x with respect to 2 provided that there exists an invertible holomorphic function e defined on such that cl x c e()