A characterization of those spaces having zero-dimensional remainders (Q1062308)

A 0-space is a completely regular Hausdorff space possessing a compactification with zero-dimensional remainder. It is well known that any rimcompact space is a 0-space, while the converse is not true. Here we present a proximal characterization of 0-spaces and characterize those open sets U of \(\beta\) X for which \(U\cap (\beta X\setminus X)\) is clopen in \(\beta\) \(X\setminus X\). This characterization is then utilized to define a relation \(\alpha\) on \({\mathcal P}(X)\). It is shown that \(\alpha\) is a proximity on X if and only if X is a 0-space. The definition of the relation \(\alpha\) is motivated by the presentation of a proximal characterization of almost rimcompact spaces - a class of spaces intermediate between the classes of rimcompact spaces and 0-spaces.