## 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.