Vol. 75, No. 2, 1978

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Decompositions of the Stone-Čech compactification which are shape equivalences

James Edgar Keesling

Vol. 75 (1978), No. 2, 455–466

Let X be a realcompact space and βX the Stone-Čech compactification of X. Let K βX X be any nondegenerate continuum. In this paper it is shown that if f(K) = Y is any map which is a shape equivalence, then f is a homeomorphism. Let X be realcompact and connected. Suppose that f(βX) = Y is a continuous map which is a shape equivalence. Then it is shown that there is a compact set K Y such that f1(K) X with f|βXf1(K) a homeomorphism onto Y K. In particular, if cX is any compactification of X and h : βX cX is the natural map induced by the identity map on X, then if h is a shape equivalence, then h is a homeomorphism. Examples and applications are given.

Mathematical Subject Classification 2000
Primary: 54C56
Secondary: 54D35, 54D60, 55D10
Received: 15 February 1977
Revised: 11 May 1977
Published: 1 April 1978
James Edgar Keesling