On zero-dimensionality of remainders of some compactications.

Abstract

A compactication of a topological space is a dense embedding of the space into a compact topological space. We study dierent methods of compactifying a topological space with the focus on zero-dimensionality of the remainder. Freudenthal compactication is known as a maximal compactication with a zero-dimensional remainder and is guaranteed to exist for rim-compact spaces. It is shown that this compactication can be characterized using proximities. In fact, there is a one-to-one correspondence between compactications and proximities and, in particular, between compactications with zero-dimensional remainder and zero-dimensional proximity. Almost rim-compact spaces are spaces that are larger than the rim-compact spaces and they are shown to also have a compactication with a zero-dimensional remainder. But these do not exhaust spaces that have a compactication with a zero-dimensional remainder, for example, recently it was found that spaces that lie between the locally compact part and its Freudenthal compactication also have a zero-dimensional remainder. It is known that the Freudenthal compactication is also perfect, we study the relationship between maximum compactications with a zero-dimensional remainder and the perfectness of these compactications.