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Simple question about topology

  1. May 15, 2006 #1

    LeonhardEuler

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    Hello, I'm reading this book and I've come to a question that has me stumped:
    I got the first part: since every set in the discrete topology is open, then the inverse image of every open set must be open.

    Its the second part that's giving me trouble. It seems to me like many functions could be continuous in the concrete topology even if they are not constant. For example, the identity function: The only open sets in the concrete toplogy are the empty set and X. The inverse image of the empty set is the empty set and the inverse image of X is X. So the inverse images of all the open sets are open. I can't seem to find any flaw in this reasoning. Can someone help me out?
     
  2. jcsd
  3. May 15, 2006 #2

    AKG

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    Assuming "concrete topology" does in fact mean that only the full and empty sets are open, your reasoning has no flaw.
     
  4. May 15, 2006 #3
    Since the question about concrete topology is a follow up to a question about discrete topology, I'd bet that the question was intended to have concrete replaced with discrete.

    SBRH
     
  5. May 15, 2006 #4

    AKG

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    Then you obviously didn't read the question.
     
  6. May 15, 2006 #5

    LeonhardEuler

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    The book says:
    So I don't think I'm misinterpreting that. I guess I just won't get hung up on this as long as I know I'm not crazy. Thank you.
     
    Last edited: May 15, 2006
  7. May 15, 2006 #6
    No, I read it.

    It is true that in the discrete topology, the only continuous functions are constant. This is not true with the concrete topology. If X is more than one point, then f(x)=x for all x in X is a continuous, non-constant function in the concrete topology.


    SBRH
     
  8. May 15, 2006 #7
    How the hell is that?
     
  9. May 15, 2006 #8

    AKG

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    Are you sure you read the question? Just before it asks about the concrete topology, it asks to prove that EVERY function X -> X in the discrete topology is continuous. Why would the next question be to show that only constant functions are continuous in the discrete topology? And you're wrong, in the discrete topology it isn't only the constant mappings that are continuous. All functions are. The original poster proved it in the first post.
     
  10. May 16, 2006 #9
    Wow! I am being totally dense. Sorry. It was one of those days.

    SBRH
     
  11. May 17, 2006 #10
    What is the name and author of the book? What page#?

    Perhaps the author refers to the fact that the only continuous maps from the concrete topology to the discrete topology are the constants.
     
  12. May 17, 2006 #11

    LeonhardEuler

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    The book is "Tensor Analysis on Manifolds" by Richard Bishop and Samuel Goldberg. The problem was on page 13 and the quote was from page 8.
     
  13. May 17, 2006 #12
    Thanks LeonhardEuler, I will attempt to look this up when I get a chance. In the meantime, is there any possibility that the authors meant "from concrete to discrete"?
     
  14. May 17, 2006 #13

    LeonhardEuler

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    I just re-checked and the wording is exactly like I have it. It is possible that the author meant that and didn't write it. The copy I have is the first printing of the book.
     
  15. May 17, 2006 #14

    matt grime

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    There is certainly a mistake somewhere. Interestingly a google search reveals the following paper citing this result with this reference:

    http://132.236.180.11/pdf/math-ph/0101032

    the article is pdf and seemingly complete nonsense (heat is a one-form??)

    It also contains links to other articles that are amusing in some sense. One of them (the one cowritten with P Bawldin asserts that:

    Well, every set in the range has an inverse image in the domain by definition of domain....
     
  16. May 17, 2006 #15

    LeonhardEuler

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    That's really weird: the one thing the author of the first article cites from the book is the very mistake this thread is about:
     
  17. May 17, 2006 #16

    matt grime

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    That's sort of why I posted it.....
     
  18. May 17, 2006 #17

    LeonhardEuler

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    Oh, I see that now. I didn't take in the full meaning of your first sentence when I first read it.
     
  19. May 17, 2006 #18
    As has already been noted, the paper by Kiehn (linked to by Matt) cites the very problem that got this thread started. Intriguingly, the actual citation (footnote #2) is to page 199 of the book by Bishop and Goldberg. LeonhardEuler, is this also an error, or is there something on page 199 related to this matter?
     
  20. May 17, 2006 #19

    LeonhardEuler

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    There is nothing about discrete or concrete topology on page 199. It's mostly about Stoke's Theorem.
     
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