POINT-SET TOPOLOGY WITH TOPICS

This textbook can be used as introduction to a general topology course at undergraduate and graduate level courses. However, many parts of this book present topological concepts that apply directly to functional analysis, which will be of interest to scholars working in those fields.

In Part I, readers are eased into the main subject matter of general topology, being presented with summaries of normed vector spaces and metric spaces. Parts II to VI form the core material contained in most Basic General Topology courses. After having worked through the most fundamental concepts of topology, the reader will be exposed to brief introductions to more specialized or advanced topics of pointset topology. These are presented in Part VII in the form of a sequence of chapters, many of which can be read or studied independently or in short sequences of two or three chapters, provided that the student has mastered the previous sections.

Chapters related to the more basic ideas of general topology are followed by a collection of 'concept review' questions, the answers to which are found in the main body of the text. These questions highlight the main concepts presented in that chapter, as well as ideas that are often overlooked when first encountered, which will help to test the students' understanding. The efforts required in answering correctly such questions will provide the student with the ability to solve more complex problems in the exercises collected at the end of each section.

Contents:

• Preface
• About the Author
• Norms and Metrics:
  • Norms on Vector Spaces
  • Metrics on Sets
• Topological Spaces: Fundamental Concepts:
  • A Topology on a Set
  • Set Closures, Interiors and Boundaries
  • Bases of Topological Spaces
  • Continuity on Topological Spaces
  • Product Spaces
  • The Quotient Topology
• Topological Spaces: Separation Axioms:
  • Separation with Open Sets
  • Separation with Continuous Functions
• Limit Points in Topological Spaces:
  • Limit Points in First Countable Spaces
  • Limit Points of Nets
  • Limit Points of Filters
• Compact Spaces and Relatives:
  • Compactness: Definition and Basic Properties
  • Countably Compact Spaces
  • Lindelöf Spaces
  • Sequentially and Feebly Compact Spaces
  • Locally Compact Spaces
  • Paracompact Topological Spaces
• The Connected Property:
  • Connected Spaces and Properties
• Topics:
  • Compactifications of Completely Regular Spaces
  • Singular Sets and Singular Compactifications
  • On C-Embeddings and Pseudocompactness
  • Realcompact Spaces
  • Perfect Functions
  • Perfect and Freudenthal Compactifications
  • Spaces Whose Elements are Sequences
  • Completing Incomplete Metric Spaces
  • The Uniform Space and the Uniform Topology
  • The Stone–Weierstrass Theorem
  • Metrizability
  • The Stone Space
  • Baire Spaces
  • The Class of F-Spaces
• Appendix
• Bibliography
• Index

Readership: This book (especially parts I-VI) is suitable for upper undergraduate and graduate level 'Introduction to General Topology' courses. Part VII covers more advanced concepts; this could be used by undergraduate students who aspire to graduate level, or in graduate classes where students develop their understanding/could be assigned certain chapters for class presentations.

Key Features:

• There are "concept review" questions at the end of each chapter to facilitate understanding and reinforce learning, the answers to which are found in the main body of the text
• There are exercise questions at the end of each section
• Instructors are able to choose from a variety of topics in this textbook, which can be assigned to graduate students for class presentations