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Topology help

  1. Dec 13, 2006 #1
    Give an example of a set X with two topologies T and S such that the identity function from (X,T) to (X,S) is continuous but not homeomorphic.

    I always struggle with these because I get overwhelmed by the generality that it has. Any ideas would be very much appreciated.
     
  2. jcsd
  3. Dec 14, 2006 #2

    AKG

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    What are the definitions of "continuous" and "homeomorphic"? Also, this post should be in the Homework Help section.
     
  4. Dec 14, 2006 #3
    i apologize i am new here....just looked for the first place that seemed appropriate


    and i assume the definitions to be the standard ones....

    continuous:
    T and S are topologies and F:X->Y; for each S open subset V of Y, f^-1(V) is a T open subset of X

    and homeomorphic:
    f is one to one, onto, continuous, and open
     
  5. Dec 14, 2006 #4

    AKG

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    Okay, and if I is the identity function from (X,T) to (X,S), V is a subset of X, what is I-1(V)? If U is a subset X, what is I(U)? Now if I is not a homeomorphism, it has to fail to have at least one of the four properties you listed under the definition of homeomorphic. If we are trying to find an example when I is continuous, then there is in fact only one of those four properties of homeomorphism that I would fail to have. Which is it?
     
  6. Dec 14, 2006 #5
    i believe youre over complicating this

    im looking for an example of a set X that has two topologies T and S that has an identity function from (X,T) to (X,S) that is continuous but not homeomorphic

    i already know that given any space X and two topologies T and S, (X,T) and (X,S) are never homeomorphic.

    im just looking for an example of this
     
  7. Dec 14, 2006 #6

    mathwonk

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    you are looking for a set with two topologies, one contained in the other. ho hum.

    try a 2 point set.
     
  8. Dec 14, 2006 #7
    using a particular point topology?

    or what?

    can you please explain

    i mean i get where youre headed but im trying to find the missing step in between
     
  9. Dec 14, 2006 #8
    nvmd...i got it....took a little longer than i hoped but i got it now

    thanks for the help
     
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