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A Guide to Topology

A Guide to Topology

Steven G. Krantz
Volume: 40
Copyright Date: 2009
Edition: 1
Pages: 120
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  • Book Info
    A Guide to Topology
    Book Description:

    A Guide to Topology is an introduction to basic topology. It covers point-set topology as well as Moore-Smith convergence and function spaces. It treats continuity, compactness, the separation axioms, connectedness, completeness, the relative topology, the quotient topology, the product topology, and all the other fundamental ideas of the subject. The book is filled with examples and illustrations. Graduate students studying for the qualifying exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful research too. Steven Krantz is well-known for his skill in expository writing and this volume confirms it. He is the author of more than 50 books, and more than 150 scholarly papers. The MAA has awarded him both the Beckenbach Book Prize and the Chauvenet Prize.

    eISBN: 978-0-88385-917-9
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. I-VI)
  2. Preface
    (pp. VII-X)
    Steven G. Krantz
  3. Table of Contents
    (pp. XI-XII)
  4. CHAPTER 1 Fundamentals
    (pp. 1-46)

    In mathematics and the physical sciences it is important to be able to compare theshapesorformsof objects. Just what do we mean by “shape”? What does it mean to say that an object has a “hole” in it? Is the hole in the center of a basketball the same as the hole in the center of a donut? Is it correct to say that a ruler and a sheet of paper have the same shape—both are, after all, rectangles? What is a rigorous and mathematical means of establishing that two objects are equivalent from the point...

  5. CHAPTER 2 Advanced Properties of Topological Spaces
    (pp. 47-68)

    We have encountered some of the ideas of this section in context in earlier parts of the book. Now we make them more formal.

    Definition 2.1.1. Let (X, 𝓤) be a topological space. We call a collection of sets 𝓼 = {Ѕα}αAabasisfor the topology 𝓤 if the collection of all unions of elements of 𝓼 equals 𝓤. Put in other words, each open setU∈ 𝓤 is the union of some elements of 𝓼.

    Example 2.1.2. LetXbe the real line with the usual topology. Then the collection of open intervals (a,b) forms a...

  6. CHAPTER 3 Moore-Smith Convergence and Nets
    (pp. 69-72)

    One of the nice features of the metric space setting is that all topological notions can be formulated in terms of sequences. Such is not the case in an arbitrary topological space. In that general setting we use the theory of nets and the associated idea of Moore-Smith convergence. That is the topic of the present chapter.

    Whereas a sequence is modeled on the natural numbers, a net is modeled on a more general object called a directed set. The notion is similar to that for sequences, but it is more abstract. We shall find good use for nets later...

  7. CHAPTER 4 Function Spaces
    (pp. 73-86)

    Many interesting examples of topological spaces arise in the context of function spaces. Function spaces are natural artifacts of analysis, and they are interesting because they are usually infinite dimensional. What does this mean?

    IfVis a vector space over the field ℝ, thenVwill have a basis.¹ If the basis has finitely many elements v₁,…,vk, then any other basis forVwill also havekelements. We callkthedimensionofV. There is also a notion of dimension for a topological space that is not necessarily a vector space. We shall not provide the details...

  8. Table of Notation
    (pp. 87-90)
  9. Glossary
    (pp. 91-98)
  10. Bibliography
    (pp. 99-102)
  11. Index
    (pp. 103-106)
  12. About the Author
    (pp. 107-107)