Almost rimcompact spaces

A 0-space is a completely regular Hausdorff space possesing a compactification with zero-dimensional remainder. Recall that a space X is called rimcompact if X has a basis of open sets with compact boundaries. It is well known that X is rimcompact if and only if X has a compactification which has a basis of open sets whose boundaries are contained in X. Thus any rimcompact space is a 0-space; the converse is not true. In this paper the class of almost rimcompact spaces is introduced and shown to be intermediate between the classes of rimcompact spaces and 0-spaces. It is shown that a space X is almost rimcompact if and only if X has a compactification in which each point of the remainder has a basis (in the compactification) of open sets whose boundaries are contained in X.