Completion of uniform and metric frames.

Abstract

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.