Locally finite nearness frames.

Abstract

The concept of a frame was introduced in the mid-sixties by Dowker and Papert. Since then frames have been extensively studied by several authors, including Banaschewski, Pultr and Baboolal to mention a few. The idea of a nearness was first introduced by H. Herrlich in 1972 and that of a nearness frame by Banaschewski in the late eighties. T. Dube made a fairly detailed study of the latter concept. The purpose of this thesis is to study the property of local finiteness and metacompactness in the setting of nearness frames. J. W. Carlson studied these ideas (including Lindelof and Pervin nearness structures) in the realm of nearness spaces. The first four chapters are a brief overview of frame theory culminating in results concerning regular, completely regular, normal and compact frames. In chapter five we provide the definitions for various nearness frames: Pervin, Lindelof , Locally Finite and Metacompact to mention a few. A particular locally finite nearness structure, denoted by µLF, is studied in detail. It is defined to be the nearness structure on a regular frame L generated by the family of all locally finite covers on the frame L. Also, a particular metacompact nearness structure, denoted by µPF, is studied in detail. It is defined to be the nearness structure on a regular frame L generated by the family of all point-finite covers of the frame L. Various theorems related the above nearness frames and these nearness structures are obtained.