Unusual topological properties
of the Nevanlinna class

Abstract: This paper considers the Nevanlinna class N of functions that are analytic in the open unit disc with bounded characteristic, and its subclass N+, the Smirnov class, both endowed with a natural complete metric in which N+ becomes a closed metric subspace of N. We prove that both N and the quotitent space N/N+ are disconnected, and show that every finite-dimensional linear subspace of N/N+ has the discrete topology (but that, nevertheless, N/N+ is not itself discrete). Furthermore, we an example of an infinite-dimensional linear subsapce of N/N+ that is discrete.

We generalize our results to other domains in the complex plane and to the polydisc and unit ball in C^n, and also consider corresponding problems for entire functions of finite order. We present a number of open questions. The most important one seeks to know if N+ is the component of the origin in N; this was answered later in the negative by James W. Roberts [The component of the origin in the Nevanlinna class, Illinois J. Math. 19 (1975), 553­559] who used the Corona Theorem to prove that the component of the origin is the closed subspace of N consisting of functions whose "denominator singular measure" has no mass points.