A characterization of 3-space by partitionings

Abstract

While it is apparent that any Euclidean n-sphere can be partitioned, it was shown only recently that any compact locally connected metric continuum is partitionable('). In this paper it is shown how a 3-sphere can be characterized in terms of its partitionings. Definitions and notation. We designate Euclidean n-space by En. The unit n-sphere with center at the origin in En+' is designated by S8. A continuum is called a simple closed curve, simple surface (or 2-sphere), or 3-sphere according as it is topologically equivalent to SI, S2, or S3. While a 3-sphere may be regarded as the set of points in E4 with coordinates (x, y, z, w) satisfying x2+y2+z2+w2= 1, we prefer to think of it as El plus a point added in such a fashion that the exterior of a cube is topologically equivalent to its interior. A simple surface C in El is called tame if there is a homeomorphism of E3 into itself that carries C into S2. If there is no such homeomorphism, C is called wild. We shall suppose that space S is metric, compact, locally connected, and connected. A partitioning of S is a collection of mutually exclusive open sets whose sum is dense in S. A sequence G1, G2, * * * of partitionings is a decreasing sequence of partitionings if Gj+1 is a refinement of Gi and the maximum of the diameters of the elements of Gi approaches 0 as i increases without limit. A partitioning is regular if each of its elements is the interior of the closure of this element. A regular partitioning G is a brick partitioning if each element of G is uniformly locally connected and the interior of the closure of the sum of each pair of elements of G is uniformly locally connected. We know that S has a decreasing sequence of brick partitionings(2). The boundary of a set A will be denoted by F(A). Characterization of a 3-sphere. A 3-sphere may be partitioned in many ways. The boundaries of some elements of a partitioning may be simple surfaces, the boundaries of others may be tori, and those of some may not even