توپولوژی عمومی (تحقیقات ریاضی مجارستان: ۹)

Preface to the English Edition Contents 1. Introduction 1.1. Sets 1.1.a. Membership 1.1.b. Subsets 1.1.c. Operations 1.1.d. Equivalence relations 1.1.e. Sets of real numbers 1.1.f. Numerical sequences 1.1.g. Countable sets 1.1.h. Well-ordering 1.1.i. Kuratowski-Zorn lemma 1.1.j. Exercises 1.2. Euclidean spaces 1.2.a. Distance 1.2.b. Convergence of a sequence of points 1.2.c. Open and closed sets 1.2.d. Dense sets 1.2.e. Theorems of Cantor, Lindelöf and Borel 1.2.f. Exercises 1.3. Metric spaces 1.3.a. Metric 1.3.b. Convergence. Open and closed sets 1.3.c. Complete metric spaces 1.3.d. Separable metric spaces 1.3.e. Distance between sets 1.3.f. Pseudo-metrics 1.3.g. Pseudo-metric spaces 1.3.h. Exercises 2. Topological spaces 2.1. The notion of topological space 2.1.a. Convergence 2.1.b. Centred systems, filter bases, filters 2.1.c. Neighbourhood structures 2.1.d. Convergence, open and closed sets in a neighbourhood space 2.1.e. Topological spaces 2.1.f. Exercises 2.2. Determination of topologies 2.2.a. Neighbourhood bases 2.2.b. Bases 2.2.c. Prescription of open or closed sets 2.2.d. Interior and closure of a set 2.2.e. The boundary of a set 2.2.f. Axiomatic remarks 2.2.g. Exercises 2.3. Comparison and restriction of topologies 2.3.a. Comparison of topologies 2.3.b. Restriction of topologies. Subspaces 2.3.c. Exercises 2.4. Convergence of filter bases 2.4.a. Insufficiency of the convergence of sequences 2.4.b. Convergence of filter bases 2.4.c. Axioms of countability 2.4.d. Examples. Metrizable spaces 2.4.e. Exercises 2.5. Separation axioms 2.5.a. Basic notions 2.5.b. T_0-spaces 2.5.c. T_1-spaces 2.5.d. T_2-spaces 2.5.e. Regular spaces 2.5.f. Normal spaces 2.5.g. Completely normal spaces 2.5.h. Exercises 2.6. Continuous mappings 2.6.a. Mappings 2.6.b. Image and inverse image of a system of sets 2.6.c. Continuous mappings 2.6.d. Homeomorphy 2.6.e. Continuous functions 2.6.f. Inverse image of topologies 2.6.g. Exercises 3. Proximity and uniform spaces 3.1. Proximity spaces 3.1.a. Proximity of a pseudo-metric space 3.1.b. Proximity spaces 3.1.c. Topology of a proximity space 3.1.d. Comparison of proximity relations 3.1.e. Restriction of proximities 3.1.f. Inverse image of proximities 3.1.g. Proximally continuous maps 3.1.h. Exercises 3.2. Uniform spaces 3.2.a. ε-Surroundings in a pseudo-metric space 3.2.b. Cartesian product of two sets 3.2.c. Uniform spaces 3.2.d. Uniformity induced by a family of pseudo-metrics 3.2.e. Proximity and topology of a uniform space 3.2.f. Comparison of uniformities 3.2.g. Restriction of uniformities 3.2.h. Inverse image of uniformities 3.2.i. Uniformly continuous maps 3.2.j. Totally bounded uniform spaces 3.2.k. Exercises 4. Completely regular spaces 4.1. Urysohn's lemma 4.1.a. Ordering in proximity and uniform spaces 4.1.b. Urysohn's lemma 4.1.c. Exercises 4.2. Completely regular spaces 4.2.a. The notion of a completely regular space 4.2.b. Families of functions 4.2.c. Inducing of topologies and proximities by function families 4.2.d. Inducing of uniformities by families of pseudo-metrics 4.2.e. Characterization by means of subbases 4.2.f. Exercises 5. Complete and compact spaces 5.1. Complete uniform spaces 5.1.a. Cauchy filter bases 5.1.b. Complete uniform spaces 5.1.c. Exercises 5.2. Compact proximity spaces 5.2.a. Compressed filter bases 5.2.b. Ultrafilters 5.2.c. Compact proximity spaces 5.2.d. Cluster points of filter bases 5.2.e. Exercises 5.3. Compact topological spaces 5.3.a. Various characterizations 5.3.b. Properties of compact spaces and sets 5.3.c. Countably compact spaces 5.3.d. Sequentially compact spaces 5.3.e. Locally compact spaces 5.3.f. Rim-compact spaces 5.3.g. Exercises 6. Extensions of spaces 6.1. Extensions of topological spaces 6.1.a. The notion of extension 6.1.b. Strict extensions 6.1.c. The Alexandroff compactification 6.1.d. \mathfrak{H}-filters 6.1.e. Wallman-type compactifications 6.1.f. Wallman compactification 6.1.g. Freudenthal compactification 6.1.h. H-closed extensions 6.1.i. Exercises 6.2. Extension of mappings 6.2.a. Extension of continuous mappings 6.2.b. Extension of uniformly continuous mappings 6.2.c. Extension of proximally continuous mappings 6.2.d. Exercises 6.3. Extensions of uniform spaces 6.3.a. Round filters 6.3.b. Extensions of a uniform space 6.3.c. Completion of a uniform space 6.3.d. Exercises 6.4. Extensions of proximity spaces 6.4.a. Extensions of a proximity space 6.4.b. Compactification of a proximity space 6.4.c. Compactifications of a completely regular space 6.4.d. Čech-Stone compactification 6.4.e. Real-compact spaces 6.4.f. Hewitt real-compactification 6.4.g. Exercises 7. Product and quotient spaces 7.1. The product of topological spaces 7.1.a. Projective generation 7.1.b. Cartesian product of sets 7.1.c. The product of topological spaces 7.1.d. The product of compact spaces 7.1.e. Embedding theorems 7.1.f. Exercises 7.2. The product of proximity spaces 7.2.a. Projective generation of proximities 7.2.b. The product of proximity spaces 7.2.c. Embedding theorems 7.2.d. Exercises 7.3. The product of uniform spaces 7.3.a. Projective generation of uniformities 7.3.b. The product of uniform spaces 7.3.c. Embedding theorems 7.3.d. Square of uniformities 7.3.e. Exercises 7.4. Quotient spaces 7.4.a. Inductive generation of topologies 7.4.b. Quotient topologies 7.4.c. Quotient spaces 7.4.d. Quotient spaces of proximity spaces 7.4.e. Quotient spaces of uniform spaces 7.4.f. Exercises 8. Paracompact spaces 8.1. Divisible spaces 8.1.a. Neighbourhoods of the diagonal 8.1.b. Multinormal spaces 8.1.c. Equicontinuous functions 8.1.d. Further characterizations 8.1.e. Exercises 8.2. Fully normal spaces 8.2.a. Refinements and star-refinements of a system of sets 8.2.b. Fully normal spaces 8.2.c. Ultracomplete spaces 8.2.d. Exercises 8.3. Paracompact spaces 8.3.a. Locally finite systems of sets 8.3.b. Paracompact spaces 8.3.c. Partitions of unity 8.3.d. Equivalent characterizations 8.3.e. Examples of paracompact spaces 8.3.f. Product theorems 8.3.g. Metacompact spaces 8.3.h. Continuous closed images of paracompact spaces 8.3.i. Countably paracompact spaces 8.3.j. Strongly paracompact spaces 8.3.k. Exercises 8.4. Metrization theorems 8.4.a. Regular and point-regular bases 8.4.b. Perfectly normal spaces 8.4.c. Metrization conditions 8.4.d. Applications 8.4.e. Metrizability of proximity spaces 8.4.f. Continuous closed images of metrizable spaces 8.4.g. Embedding into product spaces 8.4.h. Exercises 9. Baire spaces 9.1. Rare and meagre sets 9.1.a. Rare sets 9.1.b. Meagre sets 9.1.c. Exercises 9.2. Baire spaces 9.2.a. Baire spaces 9.2.b. Base-compact and hypocompact spaces 9.2.c. Product theorems 9.2.d. Applications 9.2.e. Čech-complete spaces 9.2.f. Completely metrizable spaces 9.2.g. Exercises 10. Connected spaces 10.1. Connected sets 10.1.a. Separated partitions 10.1.b. Connected sets 10.1.c. Operations 10.1.d. Components 10.1.e. Continua 10.1.f. Exercises 10.2. Locally connected spaces 10.2.a. Local connectedness 10.2.b. Operations 10.2.c. Arcs 10.2.d. Exercises 10.3. Arcwise connected spaces 10.3.a. Arcwise joined points 10.3.b. Arcwise connected sets 10.3.c. Locally arcwise connected sets 10.3.d. Chains 10.3.e. Locally connected complete metric spaces 10.3.f. Exercises 10.4. Locally connected continua 10.4.a. Covering with continua 10.4.b. Continuous images 10.4.c. Exercises 11. Topological groups 11.1. Groups 11.1.a. The notion of group 11.1.b. Examples 11.1.c. Multiplication of sets 11.1.d. Translations 11.1.e. Subgroups 11.1.f. Homomorphisms 11.1.g. Exercises 11.2. Topological groups 11.2.a. The notion of topological group 11.2.b. Neighbourhood bases of e 11.2.c. Consequences 11.2.d. Subgroups 11.2.e. Homomorphisms 11.2.f. Exercises 11.3. Complete groups 11.3.a. Admissible uniformities 11.3.b. The admissibility of left and right uniformities 11.3.c. Bilateral uniformity 11.3.d. The completion of a topological group 11.3.e. Locally compact groups 11.3.f. Exercises Literature Subject index Notations