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)

La théorie des catégories (TC) vaut tant par ses applications mathématiques que par les débats philosophiques qu'elle suscite. Elle sert à exprimer en topologie algébrique, à déduire en algèbre homologique et, en tant qu'alternative à la théorie des ensembles, à construire des objets en géométrie algébrique dans la conception de Grothendieck. Des sources non publiées montrent que Grothendieck quitta le groupe Bourbaki à l'issue d'un débat sur la TC relevant en partie de l'épistémologie, notamment quant à la réalisation ensembliste des constructions catégorielles. Nous soutenons que la TC est fondamentale, car elle traite d'opérations typiques de la mathématique de structures : d'après notre position pragmatique, la justification de la connaissance mathématique ne se fait pas par la réduction à des objets de base mais plutôt, à chaque niveau, par rapport au sens commun technique (les théories de niveau ultérieur ont pour objets les théories des objets originaux).

Made available in DSpace on 2016-06-02T20:28:22Z (GMT). No. of bitstreams: 1 DissFPM.pdf: 979275 bytes, checksum: 1b13e7e3772ecbeac26224804b180369 (MD5) Previous issue date: 2007-06-27 / Universidade Federal de Sao Carlos / In this work, we study the Artin braid group, B(n), and the confguration spaces (ordered and unordered) of a path connected manifold of dimension ¸ 2. The fundamental group of confguration space (unordered) of IR2 is identifed with the Artin braid group. This identifcation is used to conclude that the confguration space of IR2 is an Eilenberg-MacLane space of type K(B(n), 1). Therefore, it can be proved that the braid group B(n) contains no nontrivial element of the finite order. We use this fact to prove a generalization of a 2−dimensional version of the Borsuk-Ulam theorem presented by Connett [3]. / Neste trabalho, apresentamos o grupo de tranças de Artin, B(n), e os espaços de configurações (ordenado e não ordenado) de uma variedade conexa por caminhos de dimensão ¸ 2, a fim de identificar o grupo fundamental do espaço de configurações (não ordenado) de IR2 com o grupo de tranças de Artin. Usamos este fato para concluir que o espaço de configurações de IR2 é um espaço de Eilenberg-MacLane do tipo K(B(n), 1). Deste modo pode ser provado que o grupo de tranças B(n) não possui elementos não triviais de ordem finita, e usamos este fato na demonstração de uma generalização da versão bi-dimensional do teorema de Borsuk-Ulam apresentado por Connett [3].