Decompositions of the Stone-Čech compactification which are shape equivalences
James Edgar Keesling · Pacific Journal of Mathematics · 1978
Let X be a realcompact space and (3X the Stone-Cech compactification of X.Let K C (3X -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 / is a homeomorphism.Let X be realcompact and connected.Suppose that f(fiX) = Y is a continuous map which is a shape equivalence.Then it is shown that there is a compact set K C Y such that f-\K)CX with f\^X-f~\K) a homeomorphism onto Y-K.In particular, if cX is any compactification of X and h: (3X -> cX is the natural map induced by the identity map on X, then if h is a shape equivalence, then ft is a homeomorphism.Examples and applications are given.Introduction.An important question in shape theory is: What kinds of continuous mappings give shape equivalences?That is, if X