The pseudocompact extension 𝛼𝑋

Abstract

ABSTRACT. For any Tychonoff space we define aX = [ßX -vX) U X = ßX -(vX -X). We show that aX is the smallest pseudocompactification Y of X contained is ßX such that every free hyperreal z-ultrafilter on X converges in Y and is the largest pseudocompactification V of X contained in ßX such that every point in Y -X is contained in a zero set of Y which does not intersect X. A space S is defined to be a-embedded in a space X if aS C ßX. Properties of a-embeddings and its relation to u-embeddings of Blair C*-embeddings, C-embeddings, and well-embeddings are investigated. For instance, if S is a-embedded and dense in X, S is fully well-embedded (for P, R C X, where ScPCfiCX, Pis well-embedded in R) in X iff aX-aS = X -S.