Strict extensions in pointfree topology

Apfel, Gayle Renay

Strict extensions in pointfree topology

Master Thesis

2013

Publisher

University of Cape Town

Department

Department of Mathematics and Applied Mathematics

Faculty

Faculty of Science

Abstract

Extensions of spaces have been constructed and used since the 19th century, for example, to form the complex sphere from the complex plane by adding a point at in nity. Once topological spaces were invented in the 20th century, completions and compactications became important examples of extensions. Banaschewski wrote that extension problems have a """"philosophical charm"""" in that they seem to ask the question: """"What possibilities in the unknown are determined by the known?"""" Strict extensions were first defined for topological spaces by Stone. The idea was initially translated into the pointfree setting by Hong, and has since been extensively studied. Just recently, interest has been shown in studying strict extensions in the asymmetric setting of biframes, for example, by Frith and Schauerte. The intention of this dissertation is to provide a systematic and detailed exposition of strict extensions of frames and nearness frames, which can be used as a reference on this topic. For instance, someone interested in pursuing strict extensions of biframes might obtain the relevant background from reading this text, although the topic of strict extensions of biframes itself will not be discussed here.

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