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Any good problem book on General Topology

  1. Dec 29, 2009 #1
    I found it is hard for myself to follow the book on general topology by willard, since there are too many abstract definitions with too few examples to help me to establish these terms. I am wondering if there is any good problem book with sufficient problems that would help to make abstract concepts more concrete. Any suggestion will be appreciated.
  2. jcsd
  3. Dec 29, 2009 #2
    I don't know about other books, but with the majority of examples, you're really just finding ways to describe plain subsets R^n. Want a topological space? R^n. Want a compact space? A closed sphere, box, or any other shape with finite volume. Want a homeomorphism? Morph a sphere into a box. Connected spaces? R^n. Disconnected spaces? A disjoint box and sphere. Product spaces? The pairs of points on a disjoint sphere and box. Quotient spaces? A sphere where opposite points are considered equal.

    For the most part, books like that one will take the same darned examples and use them over and over again. There are more exotic spaces with less intuitive properties (such as L^infinity), but even those examples tend to get reused in the book.
  4. Dec 29, 2009 #3
    the thing is that some times it is hard to follow the book, since the problem provided as exercises have no answer, so it is hard to judge by myself whether I have the correct answer or not, so at least a solution manual for any book on general topology should be helpful, so is there any book together with solution manual available?
  5. Dec 30, 2009 #4
    These two are reasonable for problems:

    Topology Problem Solver (Problem Solvers) (Paperback) by REA

    and of course,

    Schaum's Outline of General Topology (Paperback)

    For insight:
    As far as textbooks, my favorite is the trilogy by John M Lee starting with:
    Introduction to Topological Manifolds. It's beginning graduate level but is par excellence on motivational insight. A great undergraduate text for insight is:
    Topology of Surfaces (Undergraduate Texts in Mathematics) by L.Christine Kinsey
    and there are many Dover entries. For example:
    Topology: An Introduction to the Point-Set and Algebraic Areas by Donald W. Kahn
    is very approachable (more so than Willard).

    Finally the two MIT classics are the undergrad texts by Munkres and Singer & Thorpe.

    All of the above assumes a background in real analysis "There's a delta for every epsilon" right? If that's new to you, Real Analysis by Frank Morgan is a great introduction. Don't try to push brute force through point set topology without knowledge of real analysis.
    Last edited: Dec 30, 2009
  6. Dec 30, 2009 #5
    Thanks, well, I have finished both real analysis and complex analysis, so the basic ideas like compactness or connectness or continuity are clear to me. The problem books you mentioned are also the only book i found by myself.
  7. Jan 1, 2010 #6
    I like Topology by Janich. It gets the intuition across, but it doesn't have any exercises.
  8. Jan 7, 2010 #7
  9. Jan 18, 2010 #8
    why do you want to study topology? I think the general conceptions is useless in physics.if you don't use it ,you can't master it.it is just waste of time for physicists to study it. you 'd better abandon it.
  10. Jan 18, 2010 #9

    George Jones

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    Staff Emeritus
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    Gold Member

    Are you serious?

    Firstly, this a topology forum.

    Secondly, huyichen, the original poster in this thread has not mentioned physics, and might have no interest in physics. For example, for all we know, huyichen could be a pure mathematician or a student of pure mathematics who has no interest in physics.

    Thirdly, topology is used in various areas of mathematical physics, e.g. the global methods used in general relativity. I think it would be difficult to read the proofs in Hawking and Ellis with understanding without some knowledge of topology.

    For example, it's easy to show using a topological argument,


    that any compact spacetime must have closed timelike curves (time travel).
  11. Jan 18, 2010 #10
    personally I advise you to learn general topology as you need it in other areas of mathematics. usually a first course in complex analysis will give you a strong start.
  12. Jan 19, 2010 #11
    Thanks a lot!your criticism opened my vision. Now I just read books about differential geometry in physics.I am just a student.
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