From continuity to homeomorphism, compactness in domain

  1. Is this claim true? Assume that [itex]X,Y[/itex] are topological spaces, and that all closed subsets of [itex]X[/itex] are compact. Then all continuous bijections [itex]f:X\to Y[/itex] are homeomorphisms.

    It looks true on my notebook, but I don't have a reference, and I don't trust my skills. Just checking.
  2. jcsd
  3. quasar987

    quasar987 4,773
    Science Advisor
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    There is this useful theorem that says "If X is compact and Y is Hausdorff, then any continuous bijection f:X-->Y is a homeo" and the proof goes like this:

    to show: f(C) is closed for all C in X closed. Take such a C. Since X is compact, C is compact, so f(C) is compact. But compact sets in a Hausdroff and closed, QED.

    Observe that to say that all closed subsets of X are compact is equivalent to saying that X is compact. So, modulo Y being Hausdorff, your claim is the above theorem.
  4. I see. I must have used the Hausdorff assumption without noticing it.
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