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Separated topology and existence of a metric

  1. Dec 17, 2004 #1
    Can we proove that for any separated topological space, there exists a metric?

    Seratend.
     
  2. jcsd
  3. Dec 17, 2004 #2

    matt grime

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    No, I don't think so. THere is a well known theorem that states when a space is metrizable. Try googling for it.
     
  4. Dec 17, 2004 #3

    matt grime

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    Appears you must have some kind of restriction on the cardinality of some things.

    (exactly what do you mean by separated?)
     
  5. Dec 17, 2004 #4
    sorry: direct french translation.
    for any two different points (x,y) of this set, I have at least two disjoint open sets (A,B), such that x element of A and y element of B.

    And yes, I think this is theorem I have forgotten about metrizable spaces, I am searching it now again.

    Seratend
     
  6. Dec 17, 2004 #5

    matt grime

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    oh, Hausdorff.

    you need second coutable (if it is compact) so something like a product of [0,1] indexed by some very large cardinal won't be metrizable.

    see also Uhyrson's lemma
     
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