Completion and Samuel compactification of nearness and uniform frames

Abstract

It is shown that the familiar description of the completion of a uniform frame in terms of its Samuel compactification can be extended to arbitrary nearness frames. This is achieved by means of the following new notion, a variant of compactness, for regular frames: such a frame will be called near-compact if it is complete in some totally bounded nearness. This leads to a natural concept of the Samuel near-compactification for arbitrary nearness frames which is then shown to play exactly the same role in the general setting which the Samuel compactification plays for uniform frames.