On classes of compactifications which are always singular

A compactification αX of a locally compact Hausdorff space X is said to be singular if αX β X is a retract of αX. Suppose that S is a class of locally compact, noncompact Hausdorff spaces, and that K is a collection of compact Hausdorff spaces. A general question about the existence of singular compactifications is the following: For what classes S and K is it true that each compactification of X ϵ S having a remainder αX β X ϵ K is singular? In this paper we consider a collection S, which contains the zero-dimensional spaces, and prove, among other things, that in this case K can be taken to be all products of compact metric spaces. In the process we have a variant of the well known result of Sierpiński that in a separable metric space X, a closed subset A having a zero-dimensional complement is a retract of X.