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Continuous functions have closed graphs

  1. Feb 10, 2008 #1

    quasar987

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    1. The problem statement, all variables and given/known data
    How is the theorem "Continuous functions have closed graphs" proven in the setting of a general topological space? (assuming the theorem is still valid?)
     
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  3. Feb 10, 2008 #2

    quasar987

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    Actually I found a 6 months old thread in which the OP asks help to prove that a continuous map from a topological space X to a Hausdorff space Y has a closed graph

    (https://www.physicsforums.com/showthread.php?t=171861&highlight=closed+graph)

    I haven't been able to prove this version either though. I suppose it deals with sequences and the Hausdorff property is there to ensure the uniqueness of a limit, but I'm not familiar enough with sequence theory in general topological spaces to construct an argument.
     
    Last edited: Feb 10, 2008
  4. Feb 10, 2008 #3

    HallsofIvy

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    Last edited: Feb 10, 2008
  5. Feb 10, 2008 #4

    quasar987

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    I am interested in the converse to the closed graph theorem.

    The closed graph theorem says "E and F Banach, and T linear with a closed graph. Then T is continuous."

    Very restrictive. Under which conditions is the converse true? I have shown that for a map T btw metric spaces, "T continuous ==> T has a closed graph". But I'm sure it holds under weaker hypotheses!
     
  6. Feb 10, 2008 #5

    morphism

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    I just scribbled some stuff down, and I think I managed to prove that if Y isn't Hausdorff, then there exists a topological space X and a continuous map f:X->Y whose graph isn't closed. So, provided what I did was alright (and I think it is - it wasn't a very intricate argument, but based on nets), this means that saying "the graph of any continuous function from X to Y is closed" is equivalent to saying "Y is Hausdorff".

    I don't think this is very surprising. There are a few other results of a similar flavor, e.g. a space is Hausdorff iff its diagonal is closed, a Hausdorff space is metrizable iff its diagonal is a zero set, etc.
     
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