Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics
CONTENT
0. Preliminaries -- 0.1 Conglomerates, classes and sets -- 0.2 Some categorical concepts -- 0.3 Uniform structures -- 1. Topological categories -- 1.1 Definitions and examples -- 1.2 Special categorical properties of topological categories -- 1.3 Relative connectednesses and disconnectednesses in topological categories -- 2. Reflective and coreflective subcategories -- 2.1 Universal maps and adjoint functors -- 2.2 Definitions and characterization theorems of E-reflective and M-coreflective subcategories -- 2.3 E-reflective and M-coreflective hulls -- 2.4 Reflectors as composition of epireflectors -- 3. Relations between special topological categories -- 3.1 The category Near and its subcategories -- 3.2 The category P-Near and its subcategories -- 4. Cartesian closed topological categories -- 4.1 Definitions and equivalent characterizations -- 4.2 Examples -- 5. Topological functors -- 5.1 Factorization structures -- 5.2 Definitions and properties of topological functors -- 5.3 Initially structured categories -- 6. Completions -- 6.1 Initial and final completions -- 6.2 Completion of nearness spaces -- 7. Cohomology and dimension of nearness spaces -- 7.1 Cohomology theories for nearness spaces -- 7.2 Normality and dimension of nearness spaces -- 7.3 A cohomological characterization of dimension -- Appendix. Representable functors -- Exercises
