Description
The theory of function spaces endowed with the topology of point wise convergence, or Cp-theory, exists at the intersection of three important areas of mathematics: topological algebra, functional analysis, and general topology. Cp-theory has an important role in the classification and unification of heterogeneous results from each of these areas of research. Through over 500 carefully selected problems and exercises, this volume provides a self-contained introduction to Cp-theory and general topology. By systematically introducing each of the major topics in Cp-theory, this volume is designed to bring a dedicated reader from basic topological principles to the frontiers of modern research. Key features include: – A unique problem-based introduction to the theory of function spaces. – Detailed solutions to each of the presented problems and exercises. – A comprehensive bibliography reflecting the state-of-the-art in modern Cp-theory. – Numerous open problems and directions for further research. This volume can be used as a textbook for courses in both Cp-theory and general topology as well as a reference guide for specialists studying Cp-theory and related topics. This book also provides numerous topics for PhD specialization as well as a large variety of material suitable for graduate research. Detailed summary of exercise sections Preface Introduction 1.1. Basic notions of topology and function spaces 1.1.1. Topologies, axioms of separation and a glance at Cp(X) 1.1.2. Products, cardinal functions and convergence 1.1.3. Metrizability and completeness 1.1.4. Compactness type properties in function spaces 1.1.5. More on completeness. Realcompact spaces Bibliographic notes 1.2. Solutions of problems 1.001-1.500 1.3. Bonus results: some hidden statements 1.3.1. Standard spaces 1.3.2. Metrizable spaces and compact spaces 1.3.3. Properties of continuous maps 1.3.4. Covering properties, normality and open families 1.3.5. Product spaces and cardinal invariants 1.3.6. Raznoie (unclassified results) 1.4. Open problems 1.4.1. Local properties 1.4.2. Discreteness of X and completeness of Cp(X) 1.4.3. Dense subspaces 1.4.4. The Lindelf property in X and Cp(X) 1.4.5. Other covering properties 1.4.6. Mappings which involve Cp-spaces 1.4.7. Very general questions 1.4.8. Fuzzy questions 1.4.9. Nave questions 1.4.10. Raznoie (unclassified questions) 1.5. Bibliography 1.6. List of special symbols 1.7. Index
