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## Ordered spaces of continuous functions and bitopological spaces

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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