We present three criteria for compactness in the context of apartness spaces and Bishop-style constructive mathematics. Each of our three criteria can be summarised as requiring that there is a positive distance between any two disjoint closed sets. Neat locatedness and the product apartness give us three variations on this theme. We investigate how our three criteria relate to one another and to several existing compactness criteria, namely classical compactness, completeness, total boundedness, the anti-Specker property, and Diener's neat compactness.

For a fuzzy uniform space, the notion of a Cauchy prefilter, a precompact fuzzy set, a complete fuzzy set and a bounded fuzzy set are defined in such a way that these notions are good extensions of the corresponding notions for a uniform space. A theory of fuzzy uniform spaces is developed which generalises the theory of uniform spaces.

No description available.

If (X,Ƭ) is a topological space, then a quasi-uniformity U on X is compatible with Ƭ if the quasi-uniform topology, Ƭ_u = Ƭ. This paper is concerned with local properties of quasi-uniformities on a set X that are compatible with a given topology on X. Chapter II is devoted to the construction of Hausdorff completions of transitive quasi-uniform spaces that are members of the Pervin quasi-proximity class. Chapter III discusses locally complete, locally precompact, locally symmetric and locally transitive quasi-uniform spaces. Chapter IV is devoted to function spaces of quasi-uniform spaces. Chapter V and the Appendix are concerned with the topological homeomorphism groups of quasi-uniform spaces. / Ph. D.

Bibliography: pages 90-94. / Ordered spaces were introduced by Leopoldo Nachbin [1948 a, b, c, 1950, 1965]. We will be primarily concerned with completely regular ordered spaces, because they are precisely those ordered spaces which admit quasi-uniform structures. A recent and convenient study of these spaces is in the book by P. Fletcher and W.F. Lindgren [1982]. In this thesis we consider functorial quasi-uniformities over (partially) ordered spaces. The functorial methods which we use were developed by Brummer [1971, 1977, 1979, 1982] and Brummer and Hager [1984, 1987] in the context of functorial uniformities over completely regular topological spaces, and of functorial quasi-uniformities over pairwise. completely regular bitopological spaces. We obtain results which are to a large extent analogous to results in those papers. We also introduce some functors which relate our functorial quasi-uniformities to the structures studied by Brummer and others (e.g. Salbany [1984]).

No description available.

Thesis (Ph.D.)--University of Cincinnati, 2008. / Committee/Advisors: David Herron PhD (Committee Chair), David Minda PhD (Committee Member), Nageswari Shanmugalingam PhD (Committee Member). Includes bibliographical references and abstract.

The relationship between the convergence of a sequence of self mappings of a metric space and their fixed points, known as the stability (or continuity) of fixed points has been of continuing interest and widely studied in fixed point theory. In this thesis we study the stability of common fixed points in a Hausdorff uniform space whose uniformity is generated by a family of pseudometrics, by using some general notations of convergence. These results are then extended to 2-metric spaces due to S. Gähler. In addition, a well-known theorem of T. Suzuki that generalized the Banach Contraction Principle is also extended to 2-metric spaces and applied to obtain a coincidence theorem for a pair of mappings on an arbitrary set with values in a 2-metric space. Further, we prove the existence of coincidence and fixed points of Ćirić type weakly generalized contractions in metric spaces. Subsequently, the above result is utilized to discuss applications to the convergence of modified Mann and Ishikawa iterations in a convex metric space. Finally, we obtain coincidence, fixed and stationary point results for multi-valued and hybrid pairs of mappings on a metric space.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20157号 / 理博第4242号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 照井 一成, 教授 岡本 久, 教授 長谷川 真人 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In the first part of this thesis, suitable function spaces for analysis of partial differ- ential equations in unbounded domains are introduced and studied. The results are then applied in the second part on semilinear wave equation in Rd with non- linear source term and nonlinear damping. The source term is supposed to be bounded by a polynomial function with a subcritical growth. The damping term is strictly monotone and satisfying a polynomial-like growth condition. Global existence is proved using finite speed of propagation. Dissipativity in locally uni- form spaces and the existence of a locally compact attractor are then obtained after additional conditions imposed on the damping term.