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 f−1(K) ⊂ X with f|βX−f−1(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.
