Nonscattered zero-dimension remainders (Q1263831)

For a completely regular \(T_ 2\) space X, let \(\phi\) X denote the maximal compactification of X with zero-dimensional remainder \(\phi\) X-X, if it exists. Previously the authors studied \(\phi\) X-X in case X is locally compact [Proc. Am. Math. Soc. 82, 478-480 (1981; Zbl 0469.54008)]. The present paper deals with non-locally compact X for which \(\phi\) X exists, and investigates \(\phi\) X-X on being scattered or not. Let R(X) be the set of all x in X that do not possess a compact neighbourhood in X. Several characterizations are obtained for \((\phi X- X)-Cl_{\phi}R(X)\) to be non-scattered; e.g. this is the case iff each compact metric space M is an open subspace of some remainder \(\alpha_ MX-X\) of X. As a consequence, in case R(X) is compact the same condition characterizes the X for which \(\phi\) X-X is non-scattered. Other cases treated are: R(X) is locally compact; and: X is (almost) rimcompact. Some examples are presented to show the exactness of the conditions.