categorias preço empresarial
Manual grátis

# A Gentle Path into General Topology, Part 3

### Compact, Connected and Complete Spaces and Separation Axioms

páginaprincipal.livro.por Ron McCartney
89
páginaprincipal.livro.idioma:  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.
Descrição
Conteúdos

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
páginaprincipal.livro.sobre_autor