Approximating Surfaces From the Side
R. H. Bing · Annals of Mathematics · 1963
( i ) A surface is tame if it can be homeomorphically approximated from both sides [6]. (ii) A compact connected 3-manifold is topologically S3 if each simple closed curve in it lies in a topological cube in it [14]. (iii) There is a simple closed curve that pierces no disk [5]. (iv) E3 does not contain uncountably many mutually exclusive wild 2spheres [12]. (v) Each disk in a 3-manifold can be cut down to size so as to lie in a small 2-sphere-that is, for each disk D in a 3-manifold, each point p of IntD, and each neighborhood Nof p in the 3-manifold, there is a 2-sphere S in N and a subdisk D' of D such that p e IntD' c S, [11]. (vi) Each homeomorphism of a polyhedron into a triangulated 3-manifold with boundary can be approximated by a piecewise linear homeomorphism [7].