Completion and Samuel compactification of nearness and uniform frames

Abstract

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 rôle in the general setting which the Samuel compactification plays for uniform frames.

Entourages, Density, Cauchy Maps, and Completion

“…As for the completeness questions, the reader is probably aware of a rich literature concerning this topic. To name just a few: there is the pioneering Isbell paper [14], then Kříž [17], Banaschewski [2,3], Banaschewski and Pultr [6,7,8], Pultr and Tozzi [25], Banaschewski, Hong and Pultr [4], Hong and Kim [13]. Extending the topic to the non-symmetric case opened new vistas and brought more understanding to some of the aspects (Frith and Hunsaker [11], Frith, Hunsaker and Walters-Wayland [12], Kim [16]).…”