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# A Gentle Path into General Topology, Part 1

### Part 1: Metric Spaces and their Open Sets

76
Language:  English
This book is a gentle guide along a Metric Space path, past soothing Open Subset pools, leading to the tranquil valley of Topology.
Description
Content

The most important word in the title of this book is “gentle”. The prior knowledge needed is sets, functions, proof and the real numbers. Part 1 introduces metric spaces and defines open subsets by using the concept of r-open balls. A feature of the book is the frequent “Think about” sections, in which students’ are asked to reflect on what they have just been doing. Also each chapter ends with an investigation, intended to allow students to apply their knowledge in a new context. The Bookboon website contains background material and a video introduction to each chapter.

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 1
1. Metric Spaces
1. What are they?
2. How can we prove that a function d(x, y) is not a metric ?
3. More examples of metrics
4. New metrics from old
5. Metrics defined on sets of functions
6. Investigate: Try defining more metrics
7. Appendix: A proof of the triangle inequality M4 in Rn [Sutherland (1975; p.22, 23)]
• What you can expect to learn in Chapter 2
1. Open Balls and Open Subsets in a Metric Space
1. Examples of r-open balls
2. Think about mathematical proof and the value of a diagram
3. r-open balls in R with the usual Euclidean metric
4. r-open balls in other metric spaces
5. Open subsets in a Metric Space
6. The intersection of open subsets
7. Unions of open subsets
8. Strategy to decide whether or not a subset of a metric space is open
9. Metrics which have the same open subsets
10. Investigate: Equivalent metrics
11. Appendix: Limits of sequences in a metric space
• References