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Separable space definition and applications

  1. Apr 17, 2012 #1
    I read in my metric spaces book that a separable space is that which has a countable, dense subset. This definition has no intuitive meaning to me. I'm able to show if a space is dense or not, and I think I can show a space is countable. But, I'm missing the "so what?!"

    I would like to understand this concept better. Perhaps the root of the problem is my lack of understanding of the definition of dense. I know dense means the closure of the space is equal to the space (the space and its derived set). I know if a set A is dense in B, then for every element a of A, there exists a sequence bn in B such that bn approximates a (can't say bn converges to a since a not in B).

    Am I missing something about the concept of dense maybe?

    What sorts of fun things does one do with a separable space? What does a nonseparable space imply?

    Just a note - I am not familiar with topology, only real analysis and metric spaces. The way my classes were structured, we seemed to skip most concepts involving open sets :frown:, so if there's a way to help me understand without talking about open sets or open covers, that would be really appreciated.
  2. jcsd
  3. Apr 17, 2012 #2
  4. Apr 17, 2012 #3
    if the space has only an uncountable basis, you can't build metric space structure onto it. so one way of getting a countable basis is in finding a countable dense subset to attach it to.
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