Equivalent decomposition of R³

Steve Armentrout, Lloyd Lesley Lininger and Donald Vern Meyer

Vol. 24 (1968), No. 2, 205–227

DOI: 10.2140/pjm.1968.24.205

Abstract

If G is any monotone decomposition of R^a, let H_G denote the union of the nondegenerate elements of G, and let P_G denote the projection map from R^a onto the decomposition space R³∕G associated with G. Suppose that F and G are monotone decompositions of R⁸ such that each of Cl(P_F[H_F]) and Cl(P_G[H_G]) is compact and 0-dimensional. Then F and G are equivalent decompositions of R³ if and only if there is a homeomorphism h from R³∕F onto R⁸∕G such that

A necessary and sufficient condition for two decompositions to be equivalent is given. It is shown that there is a decomposition with only a countable number of nondegenerate elements which is equivalent to the dogbone decomposition, and several related results are obtained.

Mathematical Subject Classification

Primary: 54.78

Milestones

Received: 6 April 1966

Revised: 21 April 1967

Published: 1 February 1968