On the classification of noncompact surfaces

IAN RICHARDS(i) 1. Introduction.In this paper we describe a complete topo logical classification of noncompact triangulable surfaces, and give a concrete model for an arbitrary surface, similar to the classical "normal form" for compact surfaces.This classification of arbitrary surfaces depends on the well-known classification theorem for compact surfaces, and on the idea, frequently used in the theory of Riemann surfaces, of the "ideal boundary" of a surface.The ideal boundary is a totally disconnected, compact, separable space.For our purposes, we distinguish two nested closed subsets of this space, corresponding to portions of the surface which are of "infinite genus" and "infinitely nonorientable" respectively; thus our "ideal boundary" is really a nested triple of spaces.Our first result is that, with certain fairly obvious qualifications, two surfaces are homeomorphic if and only if their ideal boundaries are topologically equivalent.This was originally discovered by Kerékjártó (see Kerékjártó [5, Chapter 5]).Kerékjártó 's proof seems to contain certain gaps, so we have included an outline of a complete proof.(See in particular the remark following Proposition 3 in §3.)In addition we prove the theorem, which so far as I know is new, that conversely, every nested triple of totally disconnected, compact, separable spaces occurs as the ideal boundary of some surface.At the same time, we show that every surface may be represented as a sphere, punctured by a finite or infinite number of discs and points, with the edges of the removed discs suitably identified.Thus we get a direct generalization of the classical representation theorem for compact surfaces.2. Basic definitions.By a surface we mean a connected 2-dimensional manifold.Just as for any manifold, one can define the property of "orientability," which a given surface may or may not have.A subset A of a surface S is said to be bounded in S if its closure in S is compact.