Vol. 24, No. 2, 1968

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Equivalent decomposition of R3

Steve Armentrout, Lloyd Lesley Lininger and Donald Vern Meyer

Vol. 24 (1968), No. 2, 205–227

If G is any monotone decomposition of Ra, let HG denote the union of the nondegenerate elements of G, and let PG denote the projection map from Ra onto the decomposition space R3∕G associated with G. Suppose that F and G are monotone decompositions of R8 such that each of Cl(PF[HF]) and Cl(PG[HG]) 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

h[Cl(PF [HF])] = Cl(PG [HG ]).

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
Received: 6 April 1966
Revised: 21 April 1967
Published: 1 February 1968
Steve Armentrout
Lloyd Lesley Lininger
Donald Vern Meyer