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Homework Help: Hausdorff spaces & Continuous Mappings via Convergence.

  1. Oct 22, 2007 #1

    strangerep

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    1. The problem statement, all variables and given/known data

    A kind person is helping me to self-study some aspects of topology,
    continuity, etc. He posed the following exercise for me, which I
    can't do, but he doesn't have time to write up the full solution.

    Ex: Show that a mapping [tex]f[/tex] between Hausdorff spaces is continuous
    if and only if the sequence [tex]f(x_0), f(x_1), \dots[/tex] converges to [tex]f(x)[/tex]
    for every sequence [itex]x_0, x_1, \dots[/itex] converging to [itex]x[/itex].

    2. Relevant equations

    3. The attempt at a solution

    To show f is continuous, one must show that every open set in the range of f(x) must
    have an inverse image which is also an open set. I also know the definition
    of a Hausdorff space as a topological space in which any two points are
    contained in disjoint open sets. But that's about as far as I get. Does this
    theorem have a name that I could look up? Or can anyone give me some
    more hints about how to progress the proof?

    TIA.
     
  2. jcsd
  3. Oct 23, 2007 #2

    EnumaElish

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    Homework Helper

    From http://en.wikipedia.org/wiki/Hausdorff_spaces#Properties

    Which tells me you can start from the reals (as a special case) then generalize.
     
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