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A Gentle Path into General Topology, Part 3

Compact, Connected and Complete Spaces and Separation Axioms

Ron McCartney
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Language:  English
Children draw pictures. When they learn language, they can explain their pictures. In topology we are learning a language to help explain our analytic, algebraic and geometric mental pictures.
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A compact topological space means that, if it is contained in an infinite union of open subsets, then a finite union of those subsets would suffice.
Connected spaces have their intuitive meaning of not being a union of disjoint non-empty open subsets.
We define completeness in metric spaces by ensuring that sequences {xn} whose terms xn come ever closer together always have a limit.
The separation axioms separate pairs of points and closed subsets, using disjoint open subsets. They take us on a journey towards metric spaces again. By investigating further, students may complete the journey.

About the Author

I had a thorough training in analysis and general topology as an undergraduate, and my doctoral research was in nets and compactifications. I have taught analysis and general topology in three universities, most recently in Asia, but earlier in the UK Open University, where I was also involved in mathematics education. I enjoy helping students appreciate the precision and rigour of analysis and topology, and I continue to gain much pleasure through investigating their secrets.

  • Foreword
  • Preface
  • Introduction
  • What you can expect to learn in Chapter 5
  1. Compact Spaces
    1. Compactness of a product space
    2. Compactness in terms of closed subsets
    3. Sequential Compactness
    4. Investigations
  • What you can expect to learn in Chapter 6
  1. Connected Spaces
    1. Continuity and connectedness
    2. The product of connected spaces
    3. Connected components
    4. Investigations
  • What you can expect to learn in Chapter 7
  1. Complete Metric Spaces
    1. Sequences in a metric space
    2. Cauchy sequences and completeness
    3. Completion of a non-complete metric space
    4. Compactness and completeness
    5. Investigations
    6. Appendices 1: A proof of the completeness of the Real Numbers R
    7. Appendices 2: Constructing the completion of a non-complete metric space
  • What you can expect to learn in Chapter 8
  1. Separation Axioms
    1. T1 - spaces
    2. Hausdorff (or T2) Spaces
    3. Regular (or T3) spaces
    4. Normal (or T4) spaces
    5. Investigations
    6. Appendix: Looking back and looking forward
  • References