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A Gentle Path into General Topology, Part 2

Topological Spaces and Continuity

Ron McCartney
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101
Language:  English
“What is a topological space?” and “What does it mean to say that two topological spaces are the same?” These are the two topics explored in this part of the book.
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Content
Description

  • Foreword
  • Preface
  • Introduction
  • What you can expect to learn in Chapter 3
  1. Topological Spaces
    1. What are they ?
    2. Ordering the topologies on a set
    3. The relationship between metric spaces and topological spaces
    4. Building a topology on a set
    5. Forming topologies from the ground up
    6. Subspaces
    7. Closed sets and closure of a set
    8. Investigate Neighbourhoods
    9. Appendix : In a metric space sequences may be used to define closure
  • What you can expect to learn in Chapter 4
  1. Continuity
    1. What is it ?
    2. Continuity at every point of the domain of a function
    3. Spaces which are topologically the same
    4. Shapes in the plane
    5. The Product Topology
    6. Extending the definition of product topology
    7. New topological spaces from old
    8. Investigate : More on quotient spaces
    9. Appendix : The Importance of Continuity in Analysis and Topology
  • References

In metric spaces finite intersections of open subsets, all unions of open subsets, the whole space and the empty set are always open. A topological space is a non-empty set with a collection of subsets, which possesses those properties.
  Intuitively, we say that a function f is continuous at a particular element b, if images f(x) may be made as close as we like to the image f(b), for all x sufficiently close to b. The topological interpretation is: f is continuous at b if, for every open subset H containing f(b), we can always find an open subset G, whose image, f(G) is contained in H.

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.