This thesis is divided into two parts: Ordered spaces of Continuous Functions and
the algebras associated with the topology of pointwise convergence of the associated
construct, and Strictly completely regular bitopological spaces.
The Motivation for part of the first part (Chapters 2, 3 and 4) comes from the
recent study of function spaces for bitopological spaces in [44] and [45]. In these
papers we see a clear generalisation of classical results in function spaces ( [14] and
[55]) to bi-topological spaces. The well known definitions of the pointwise topology and
the compact open topology in function spaces are generalized to bitopological spaces,
and then familiar results such as Arens' theorem are generalised. We will use the same
approach in chapters 2, 3 and 4 to formulate analogous definitions in the setting of
ordered spaces. Well known results, including Arens' theorem, are also generalised
to ordered spaces. In these chapters we will also compare function spaces in the
category of topological spaces and continuous functions, the category of bi topological
spaces and bicontinuous functions, and the category of ordered topological spaces and
continuous order-preserving functions. This work has resulted in the publication of
[30] and [31].
Continuing our study of Function Spaces, we oonsider in Chapters 5 and 6 some
Categorical aspects of the construction, motivated by a series of papers which includes
[39], [40], [41] and [50]. In these papers the Eilenberg-Moore Category of algebras of
the monad induced by the Hom-functor on the categories of sets and categories of
topological spaces are classified. Instead of looking at the whole product topology we
will restrict ourselves to the pointwise topology and give examples of the EilenbergMoore Algebras arising from this restriction. We first start by way of motivation, with
the discussion of the monad when the range space is the real line with the usual topology.
We then restrict our range space to the two point Sierpinski space, with the aim
of discovering a topological analogue of the well known characterization of Frames as
the Eilenberg-Moore Category of algebras associated with the Hom-F\mctor of maps
into the Sierpinski space [11]. In this case the order structure features prominently, resulting in the category Frames with a special property called "balanced" and Frame
homomorphisms as the Eilenberg-Moore category of M-algebras. This has resulted
in [34].
The Motivation for the second part comes from [20] and [15]. In [20], J. D. Lawson
introduced the notion of strict complete regularity in ordered spaces. A detailed study
of this notion was done by H-P. A. Kiinzi in [15]. We shall introduce an analogous
notion for bitopological spaces, and then shall also compare the two notions in the categories
of bi topological spaces and bicontinuous functions, and of ordered topological
spaces and continuous order-preserving functions via the natural functors considered
in the previous chapters. We further study the Stone-Cech bicompactification and
Stone-Cech ordered compactification in the two categories. This has resulted in [32] and [33] / Mathematical Sciences / D. Phil. (Mathematics)
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:unisa/oai:umkn-dsp01.int.unisa.ac.za:10500/17559 |
| Date | 11 1900 |
| Creators | Nailana, Koena Rufus |
| Contributors | Salbany, Sergio, Alderton, Ian William, 1952- |
| Source Sets | South African National ETD Portal |
| Detected Language | English |
| Type | Thesis |
| Format | 1 online resource (vi, 122 leaves) |
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