Approximating Surfaces With Polyhedral Ones

Abstract

If J is a simple closed curve in the plane E2, the Schoenflies Theorem says that there is a homeomorphism of E2 onto itself that takes J onto a circle. The theorem does not generalize directly to E3 because there is a simple S in E3 such that there is no homeomorphism of E3 onto itself that takes S onto the of a sphere. However, if S is polyhedral, there is a homeomorphism of E3 onto itself that takes S onto the of a sphere [1]. Hence, polyhedral surfaces are imbedded in E3 in a simpler fashion than are some other surfaces. The intersection of two polyhedral surfaces is much less complicated than the intersection of some other surfaces. For this reason, mathematicians sometimes impose the additional condition of being polyhedral on some of the surfaces they use. Applications of the results of this paper show that this extra condition is not always necessary. Harrold gives several applications of the approximation theorem for surfaces (Theorem 7) of this paper in [7]. I make extensive use of it in showing that there is a simple closed curve in E3 which pierces no disk [4]. In a subsequent paper I shall show that an extension of this theorem to topological 2-complexes can be used to give alternate proofs of the theorems due to Moise to the effect that each 3-manifold can be triangulated [9] (also a 3-manifold with boundary can be triangulated [2]) and that each homeomorphism of one 3-manifold with boundary into another can be approximated arbitrarily closely by a piecewise linear homeomorphism [8]. In this paper we consider the extent to which arbitrary surfaces can be approximated by polyhedral ones. We show that each can be homeomorphically approximated arbitrarily closely by one that is locally polyhedral. Therefore, it can be approximated by a polyhedron if it is compact [Lemma 1 of 2]. A simple is a continuum topologically equivalent to the of a sphere. We use the term surface synonomously with 2-manifold with boundary. An n-manifold is a separable metric space K such that each point of K lies in an open set that is topologically equivalent to En (Euclidean n-space). An n-manifold with boundary is a separable metric space K such that each point of K lies in an open set N such that I7 is topologically equivalent to In (cube in En). We note that an n-manifold is an n-manifold with boundary but not conversely. The term boundary is used in two senses. In the point set sense we say that the boundary of a set X is the intersection of X (the closure of X) and the closure