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. / Thesis (M.Sc.)-University of Durban-Westvile, 1998.

The term "frame" was introduced by C H Dowker, who studied them in a long series of joint papers with D Papert Strauss. J R Isbell , in a path breaking paper [1972] pointed out the need to introduce separate terminology for the opposite of the category of Frames and coined the term "locale". He was the progenitor of the idea that the category of Locales is actually more convenient in many ways than the category of Frames. In fact, this proves to be the case in one of the approaches adopted in this thesis. Sublocales (quotient frames) have been studied by several authors, notably Dowker and Papert [1966] and Isbell [1972]. The term "sublocale" is due to Isbell, who also used "part " to mean approximately the same thing. The use of nuclei as a tool for studying sublocales (as is used in this thesis) and the term "nucleus" itself was initiated by H Simmons [1978] and his student D Macnab [1981]. Uniform spaces were introduced by Weil [1937]. Isbell [1958] studied algebras of uniformly continuous functions on uniform spaces. In this thesis, we introduce the concept of a uniform frame (locale) which has attracted much interest recently and here too Isbell [1972] has some results of interest. The notion of a metric frame was introduced by A Pultr [1984]. The main aim of his paper [11] was to prove metrization theorems for pointless uniformities. This thesis focuses on the construction of completions in Uniform Frames and Metric Frames. Isbell [6] showed the existence of completions using a frame of certain filters. We describe the completion of a frame L as a quotient of the uniformly regular ideals of L, as expounded by Banaschewski and Pultr[3]. Then we give a substantially more elegant construction of the completion of a uniform frame (locale) as a suitable quotient of the frame of all downsets of L. This approach is attributable to Kriz[9]. Finally, we show that every metric frame has a unique completion, as outlined by Banaschewski and Pultr[4]. In the main, this thesis is a standard exposition of known, but scattered material. Throughout the thesis, choice principles such as C.D.C (Countable Dependent Choice) are used and generally without mention. The treatment of category theory (which is used freely throughout this thesis) is not self-contained. Numbers in brackets refer to the bibliography at the end of the thesis. We will use 0 to indicate the end of proofs of lemmas, theorems and propositions. Chapter 1 covers some basic definitions on frames , which will be utilized in subsequent chapters. We will verify whatever we need in an endeavour to enhance clarity. We define the categories, Frm of frames and frame homomorphisms, and Lac the category of locales and frame morphisms. Then we explicate the adjoint situation that exists between Frm and Top , the category of topological spaces and continuous functions. This is followed by an introduction to the categories, RegFrm of all regular frames and frame homomorphisms, and KRegFrm the category of compact regular frames and their homomorphisms. We then present the proofs of two very important lemmas in these categories. Finally, we define the compactification of and a congruence on a frame. In Chapter 2 we recall some basic definitions of covers, refinements and star refinements of covers. We introduce the notion of a uniform frame and define certain mappings (morphisms) between uniform frames (locales) . In the terminology of Banaschewski and Kriz [9] we define a complete uniform frame and the completion of a uniform frame. The aim of Chapter 3 is twofold : first, to construct the compact regular coreflection of uniform frames , that is, the frame counterpart of the Samuel Compactification of uniform spaces [12] , and then to use it for a description of the completion of a uniform frame as an alternative to that previously given by Isbell[6]. The main purpose of Chapter 4 is to provide another description of uniform completion in frames (locales), which is in fact even more straightforward than the original topological construction. It simply consists of writing down generators and defining relations. We provide a detailed examination of the main result in this section, that is, a uniform frame L is complete of each uniform embedding f : (M,UM) -t (L,UL) is closed, where UM and UL denote the uniformities on the frames M and L respectively. Finally, in Chapter 5, we introduce the notions of a metric diameter and a metric frame. Using the fact that every metric frame is a uniform frame and hence has a uniform completion, we show that every metric frame L has a unique completion : CL - L. / Thesis (M.Sc.)-University of Durban-Westville, 1996.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 219-232). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph.D.)--York University, 2008. Graduate Programme in Communication and Culture. / Typescript. Includes bibliographical references (leaves 400-427). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR39042

Thesis (M.A.)--University of Missouri-Columbia, 2005. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (July 11, 2006) Includes bibliographical references.

Thesis (Ph. D.)--University of Missouri-Columbia, 2007. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on December 14, 2007) Vita. Includes bibliographical references.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg. in fulfilment of the requirements tor the degree of Master of Science / This thesis develops the theoretical basis of the qualitative frame based modelling technique, a paradigm recently proposed by Starfield for the modelling of ecosystems with a multiplicity of stable states. This technique is a refinement of the State-and- Transition conceptual model of Westoby et al which involves the division of the ecosystem dynamics into a catelog of stable 'states' and a suite of transitions between these states. The frame models of Starfield associate with each stable configuration of the ecosystem a qualitative rule based model for the key processes in that stable configuration. The aims of this thesis are the following, 1. A rigorous definition of frame modelling of dynamic ecosystems is proposed, and this theoretical foundation is used to demonstrate that qualitative frame models may be used to mode! dynamic ecosystems to an arbitrary accuracy. 2. The development of implementation software. A qualitative rule based frame modelling environment is presented. and a specification for an improved environment is proposed based on the theoretical work. / Andrew Chakane 2019

We present the concept of a frame and the related notion of spatiality. We consider the classical separation axioms in the frame setting and relate these to frame covering properties. Finally, a determination of which covering properties and separation axioms imply spatiality of a frame is made. / Master of Science