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Topology for Analysis

  1. May 2, 2012 #1
    Where should I start studying topology for analysis? I'm completely new to the subject of topology, and I found there are different areas of topology, but my concern is the one that mostly maps to analysis concepts. Besides I know Munkre's Topology is the standard, but I'm not specializing in the subject and not planning to spend much on the standard text for now.
    There are many Dovers but which one?

    Thanks.
     
  2. jcsd
  3. May 2, 2012 #2
    "Topology for Analysis" By Wilansky, or "General Topology" By Kelley are both from dover
     
  4. May 2, 2012 #3

    lavinia

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    In many books on Complex Analysis, topology is developed as a tool to understand analytic functions. Basic point set topology, the Jordan curve theorem, elementary ideas of homotopy and homology and the idea of a Riemann surface are all developed.
     
  5. May 2, 2012 #4
    So if I'm not specializing in topology I should just count on the preliminary materials presented in analysis? As I understand topology explains main concepts in analysis but in a dimensionless generic way, such as continuity, convergence, smoothness...etc.
     
  6. May 2, 2012 #5

    micromass

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    Just get a good real/complex analysis book which spends some chapters on topology. There are many good such books. However, if you want more than a basic working knowledge, then you'd have to get a topology book.
     
  7. May 2, 2012 #6
    How much helpful knowing topology in depth? any applicable areas in pure math?
     
  8. May 2, 2012 #7
    Depends on how much depth you are thinking of. Almost all areas of pure math will require you to know at least the level of topology one would gain from a course based on Munkres at some point (for instance separation axioms, various forms of connectedness, various forms of compactness, Urysohn's lemma, Tietze's extension theorem, Baire's category theorem, and the fundamental group were all used in a graduate course on functional analysis I recently took). However if you do not really like topology yet you can postpone it a bit till you feel the need or urge to study topology by itself.

    EDIT: And yes for the first couple of analysis courses you can likely expect a couple of lectures on the required topology. You usually will not need a topology prerequisite for the first classes. Around the time you get into functional analysis, operator algebras, non-commutative geometry or some other graduate level topic you may start seeing a course in topology as a prerequisite, but this obviously depends on the course in question. It sounds like this is still some way down the road. Don't worry about it yet if you are only about to learn real/complex analysis.
     
    Last edited: May 2, 2012
  9. May 2, 2012 #8

    lavinia

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    I would pick the topology up along the way. Complex analysis is a great beginning.

    I studied point set topology separately and never found it useful by itself.
     
  10. May 2, 2012 #9

    mathwonk

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    I felt the same about point set topology, but I might make a few provisos. It may be that I did get familiarity with a basic language which I then always took for granted. Thats the good part. The bad part was that I did not examine enough exampkles, just memorized a lot of abstract proofs of very abstract theorems like tychonoff and baire, stuff like that.

    actually i learned later from errett bishop i think, that baire category is a very strong and useful theorem.

    i would say the first part of that munkres book is the part analysts would use.


    kelley says in his introduction i believe that it was only with difficulty that his friends prevented him from titling his book "what every young analyst should know". I learned some basic stuff from there, but it is pretty abstract.

    the topology in the first 80-130 pages of dieudonne's Foundations of modern analysis, is obviously slanted toward analysis. It may be hard to read, but is very rewarding.
     
  11. May 4, 2012 #10
    marsdens elementary classical analysis first few chapters cover a good amount of topology commonly used in analysis arguements. however it is not enough for differential geometry. for that I might suggest Lee's topological manifolds and smooth manifolds together. It is easier in my opinion to learn topology through a geometric model
     
  12. May 4, 2012 #11
    That's what I'm afraid of find myself jumping from one book to another to grab information needed for different areas of analysis. What would be the comprehensive guide on topology for all kinds of analysis, including multidimensional, manifolds, and complex?
     
  13. May 4, 2012 #12

    micromass

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    In your OP you say you don't want a standard text since you don't want to spend much time on it, and now you say that you want a comprehensive guide. What will it be??

    Perhaps you should say what your eventual goal is. Why do you wish to study all this math?? Perhaps then we'll be able to give you more information.
     
  14. May 4, 2012 #13
    Yeah my OP is not clear. I meant to say that if there's no need to study topology in depth in order to do analysis, including multidimensional analysis, then this would saves me a lot of time and money on a single text. But I would prefer one text to refer over multiple texts if I have to.
     
  15. May 4, 2012 #14

    mathwonk

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    There are several criteria for a book, that it should cover the most essential material, but equally important, that it be a book at the right level for you to understand.


    Here is an outstanding book, mentioned above, if you can benefit from it:

    http://www.abebooks.com/servlet/SearchResults?an=dieudonne&kn=foundations&x=47&y=11

    But the best way to find out is to peruse it, and others suggested here, in a library before buying it. However, I would recommend this book to anyone seriously interested in mathematics. And i doesn't get old fast, I still have my copy and still look at it from time to time.


    here is the table of contents:

    https://www.amazon.com/Treatise-Ana...=sr_1_3?s=books&ie=UTF8&qid=1336151346&sr=1-3
     
    Last edited: May 4, 2012
  16. May 4, 2012 #15
    The books I have cited are standard books on real analysis, smooth manifolds and topological manifolds. If your gradual goal is to do analysis and geometry, you would benefit from reading those and you will have to learn those anyway. That would enable you to learn topology by applying it to analysis and geometry examples. Otherwise you will have to find a book on abstract topology, study it through first and then move to manifolds and analysis. I personally can not remember abstract theorems, even if I study their proofs in detail, without applying them to a geometrical or physical example.

    Ofcourse that would make it faster and easier for you to study analysis and geometry afterwards but overall I think it wouldn't be a difference.
     
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