Topological Spaces

Topological Spaces focuses on the applications of the theory of topological spaces to the different branches of mathematics.

The book first offers information on elementary principles, topological spaces, and compactness and connectedness. Discussions focus on locally compact spaces, local connectedness, fundamental concepts and their reformulations, lattice of topologies, axioms of separation, fundamental concepts of set theory, and ordered sets and lattices. The manuscript then ponders on mappings and extensions and characterization of topological spaces, including completely regular spaces, transference of topologies, Wallman compactification, and embeddings.

The publication takes a look at metric and uniform spaces and applications of topological groups. Topics include the Stone-Weierstrass Approximation Theorem, extensions and completions of topological groups, topological rings and fields, extension and completion of uniform spaces, uniform continuity and uniform convergence, metric spaces, and metritization.

The text is a valuable reference for mathematicians and researchers interested in the study of topological spaces.

Contenu

Preface

Introduction

I. Elementary Principles

1. Fundamental Concepts of Set Theory

2. Ordered Sets and Lattices

3. Filters

II. Topological Spaces

4. Topology and Limits

5. Fundamental Concepts and Their Reformulations

6. The Order Topology and Metric Spaces

7. The Lattice of Topologies

8. Axioms of Separation

9. Subspaces

10. Cardinality Conditions

III. Compactness and Connectedness

11. Compactness

12. Locally Compact Spaces

13. Paracompact Spaces

14. Connectedness

15. Local Connectedness

IV. Mappings

16. Topological Maps, Homeomorphisms

17. Continuous Mappings

18. Completely Regular Spaces

19. Properties of Continuous Mappings

20. Transference of Topologies

21. Quotient Spaces

22. Product Spaces

23. Spaces of Mappings

24. Sums of Spaces

V. Extensions and Characterization of Topological Spaces

25. Extensions of Topological Spaces

26. The Wallman Compactification

27. The Stone-Cech Compactification

28. Embeddings

29. Representation Theorems

VI. Metric and Uniform Spaces

30. Metric Spaces

31. Metrization

32. Extensions and Completions of Metric Spaces

33. Uniform Structures

34. Uniform Continuity and Uniform Convergence

35. Extension and Completion of Uniform Spaces

VII. Topological Groups. Applications

36. Topological Groups

37. Extensions and Completions of Topological Groups

38. Topological Rings and Fields

39. Topological Vector Spaces. The Stone-Weierstrass Approximation Theorem

40. Inductive and Projective Limits

Notations Employed

Bibliography

Subject Index