<p>The main purpose of this thesis is to study the interplay between relational structures and topology , and to portray pairings in terms of some finite poset models and order preserving maps. We show the interrelations between the categories of topological spaces, closure spaces and relational structures. We study the 4-point non-Hausdorff model S4 weakly homotopy equivalent to the circle S1. We study pairings of some objects in the category of relational structures similar to the multiplication S4 x S4- S4 S4 fails to be order preserving for posets. Nevertheless, applying a single barycentric subdivision on S4 to get S8, an 8-point model of the circle enables us to define an order preserving poset map S8 x S8- S4. Restricted to the axes, this map yields weak homotopy equivalences S8 x S8, we obtain a version of the Hopf map S8 x S8s - SS4. This model of the Hopf map is in fact a map of non-Hausdorff double map cylinders.</p>

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Thesis(M.S.)--Auburn University, 2006. / Abstract. Vita. Includes bibliographic references (p.22).

Thesis (M. Sc. (Mathematics))--Rhodes University, 2004.

Thesis (Ph.D.)--Bowling Green State University, 2009. / Document formatted into pages; contains vi, 119 p. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1967. / Typescript. Vita. Includes bibliographical references.

Magister Scientiae - MSc / The main purpose of this thesis is to study the interplay between relational structures and topology , and to portray pairings in terms of some finite poset models and order preserving maps. We show the interrelations between the categories of topological spaces, closure spaces and relational structures. We study the 4-point non-Hausdorff model S4 weakly homotopy equivalent to the circle S1. We study pairings of some objects in the category of relational structures similar to the multiplication S4 x S4- S4 S4 fails to be order preserving for posets. Nevertheless, applying a single barycentric subdivision on S4 to get S8, an 8-point model of the circle enables us to define an order preserving poset map S8 x S8- S4. Restricted to the axes, this map yields weak homotopy equivalences S8 x S8, we obtain a version of the Hopf map S8 x S8s - SS4. This model of the Hopf map is in fact a map of non-Hausdorff double map cylinders. / South Africa

Aspects of fuzzy vector spaces and fuzzy groups are investigated, including linear independence, basis, dimension, group order, finitely generated groups and cyclic groups. It was necessary to consider cardinality of fuzzy sets and related issues, which included a question of ways in which to define functions between fuzzy sets. Among the results proved, are the additivity property of dimension for fuzzy vector spaces, Lagrange's Theorem for fuzzy groups ( the existing version of this theorem does not take fuzziness into account at all), a compactness property of finitely generated fuzzy groups and an extension of an earlier result on the order-homomorphisms. An open question is posed with regard to the existence of a basis for an arbitrary fuzzy vector space.

The objective of this thesis is to develop certain aspects of the theory of (L,M)-fuzzy topological spaces, where L and M are complete lattices (with additional conditions when necessary). We obtain results which are to a large extent analogous to results given in a series of papers of Šostak (where L = M = [0,1]) but not necessarily with analogous proofs. Often, our generalizations require a variety of techniques from lattice theory e.g. from continuity or complete distributive lattices.

The objective of this thesis is a survey of crisp and fuzzy sober topological spaces. We begin by examining sobriety of crisp topological spaces. We then extend this to the L- topological case and obtain analogous results and characterizations to those of the crisp case. We then brie y examine semi-sobriety of (L;M)-topological spaces.