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Continuous Maps and Products

  1. Dec 26, 2012 #1
    Let f: A -> B and g: C -> D be continuous functions.

    Define h: A x C -> B x D by the equation

    h(a,c)=(f(a),g(c)).​

    Show h is continuous.


    A few weeks ago I completed this exercise. Now, I am working on a problem that would be almost too easy if the converse of the above claim were true. I had trouble trying to construct a counterexample; so I tried to prove it.

    Suppose h is continous. Let U be open in B. Then U x D is open in B x D, and by our assumption, h-1(U x D) is open in A x C. Since h-1(U x D) = f-1(U) x g-1(D), f-1(U) is open and f is continuous.

    Does f-1(U) x g-1(D) being open in the product space imply f-1(U) is open in A?
     
  2. jcsd
  3. Dec 26, 2012 #2

    micromass

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    Are projections open maps?
     
  4. Dec 26, 2012 #3
    Yes.

    Forgive my naivety, but how come the exercise isn't presented as an "if and only if" statement?

    Perhaps I should examine these problems more closely. I originally completed this problem rather quickly; I then moved on without a second thought about if the converse is true. I should pay more attention.
     
  5. Dec 26, 2012 #4

    micromass

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    The converse seems much less interesting, so I can understand why it is not an exercise. In either case, thinking about the converse is always a good practice.
     
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