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Separable metric space?

  1. May 25, 2007 #1
    1. The problem statement, all variables and given/known data
    'In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space.'


    Let (X,d) be a metric space. If X is countable than it immediately satisfies being a separable metric space? Because just choose X itself as the subset. The closure of X must be X. Hence there exists a countable dense subset, namely X itself.

    3. The attempt at a solution
    Is this correct?

    Or they referring to proper subsets only?
  2. jcsd
  3. May 26, 2007 #2


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    If they meant proper, they'd say it. Admitting only proper subsets is equivalent to excluding countable sets. There's no good reason for doing that. Also, that article gives an example of a countable space being separable.
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