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A new foundation of Topology, summarized under the name Convenient Topology, is considered such that several deficiencies of topological and uniform spaces are remedied. This does not mean that these spaces are superfluous. It means exactly that a better framework for handling problems of a topological nature is used. In this setting semiuniform convergence spaces play an essential role. They include not only convergence structures such as topological structures and limit space structures, but also uniform convergence structures such as uniform structures and uniform limit space structures, and they are suitable for studying continuity, Cauchy continuity and uniform continuity as well as convergence structures in function spaces, e.g. simple convergence, continuous convergence and uniform convergence. Various interesting results are presented which cannot be obtained by using topological or uniform spaces in the usual context. The text is self-contained with the exception of the last chapter, where the intuitive concept of nearness is incorporated in Convenient Topology (there exist already excellent expositions on nearness spaces).
Table des matières
0. Preliminaries.- 0.1 Set theoretical concepts.- 0.2 Topological structures.- 0.3 Some categorical concepts.- 1. Topological constructs.- 1.1 Definition and examples.- 1.2 Special categorical properties of topological constructs.- 2. Reflections and coreflections.- 2.1 Universal maps and adjoint functors.- 2.2 Characterization theorems of ?-reflective and M-coreflective subcategories.- 2.3 Examples of bireflections and bicoreflections.- 3. Topological universes.- 3.1 Cartesian closed topological constructs.- 3.2 Extensional topological constructs.- 3.3 Strong topological universes.- 4. Completions of semiuniform convergence spaces.- 4.1 Completion of uniform spaces.- 4.2 Regular completion of semiuniform convergence spaces.- 4.3 Applications to compactifications.- 4.4 The simple completion and the Wyler completion.- 5. Connectedness properties.- 5.1 Connectednesses.- 5.2 Disconnectednesses and their relations to connectednesses.- 5.3 Local ?-connectedness.- 6. Function spaces.- 6.1 Simple convergence, uniform convergence and continuous convergence in the realm of classical General Topology.- 6.2 Local compactness and local precompactness in semiuniform convergence spaces.- 6.3 Precompactness and compactness in the natural function spaces of the construct of semiuniform convergence spaces.- 7. Relations between semiuniform convergence spaces and merotopic spaces (including nearness spaces).- 7.1 An alternative description of filter spaces in the realm of merotopic spaces.- 7.2 Subtopological spaces.- 7.3 Complete regularity and normality.- 7.4 Paracompactness and dimension.- 7.5 Subcompact and sub-(compact Hausdorff) spaces.- Appendix. Some algebraically topological aspects in the realm of Convenient Topology.- A.l Cohomology for filter spaces.- A.2 Path connectednessand fundamental groups for limit spaces.- Exercises.- Implication scheme for various SUConv-invariants.- Table: Preservation properties of some SUConv-invariants.- Diagram of relations between various subconstructs of SUConv (including their relations to merotopic and nearness spaces).- List of axioms.- List of symbols.
Résumé
A new foundation of Topology, summarized under the name Convenient Topology, is considered such that several deficiencies of topological and uniform spaces are remedied. This does not mean that these spaces are superfluous. It means exactly that a better framework for handling problems of a topological nature is used. In this setting semiuniform convergence spaces play an essential role. They include not only convergence structures such as topological structures and limit space structures, but also uniform convergence structures such as uniform structures and uniform limit space structures, and they are suitable for studying continuity, Cauchy continuity and uniform continuity as well as convergence structures in function spaces, e.g. simple convergence, continuous convergence and uniform convergence. Various interesting results are presented which cannot be obtained by using topological or uniform spaces in the usual context. The text is self-contained with the exception of the last chapter, where the intuitive concept of nearness is incorporated in Convenient Topology (there exist already excellent expositions on nearness spaces).
