Locally Tame Sets are Tame

Abstract

In this paper we prove some theorems concerning 3-manifolds with boundaries. Any such set can be triangulated (Theorem 5). Furthermore, any locally tame closed subset of such a set is tame (Theorem 9). Working independently, Moise has subsequently obtained these results and is including his proofs along with other results in [5]. Much of the terminology used in this paper resembles that used in [4]. An n-manifold is a separable metric space K such that each point p of K lies in an open set that is topologically equivalent to En (Euclidean n-space). An n-manifold with is a separable metric space K such that each point p of K lies in an open set N such that N is topologically equivalent to I' (a cube plus its interior in En). We note that an n-manifold is an n-manifold with but not conversely. The term boundary is used in two senses. In the point set sense we say that the of a set X is the intersection of the closure of X and the closure of the complement of X. However, if K is an n-manifold with boundary, we use Bd K to denote the points of K which do not have a neighborhood topologically equivalent to En. For example, if in a triangulated 3-manifold M with boundary, St(v) represents the star of a vertex v (the sum of all simplexes having a vertex at v) then Bd St(v) will be larger than the point set of St(v) if v belongs to Bd M. In this paper we always use Bd(K) to denote the set of points of K which do not have a neighborhood topologically equivalent to En. The interior of an n-manifold K with is K Bd K and is denoted by Int K. A set K can be triangulated if there is a homeomorphism h carrying a geometric complex C onto K. (We suppose that complexes are locally finite.) Let T(K) denote the collection of all subsets o of K such that ais the image under h of a simplex of C. If (a, xi, a2 x2, ... , ai xi) represents a point p of C