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Am I right?

  1. Aug 22, 2009 #1
    I had this thought:
    Every interval (a,b) of the real line is a closed and bounded non-compact topological space.

    Is this correct?
  2. jcsd
  3. Aug 23, 2009 #2
    Interval (a,b) is not closed.
  4. Aug 23, 2009 #3
    Closed and non-compact in what space? Trivially, any topological space is closed in itself (which is why we don't really say that something is a "closed topological space"), is this what you mean?

    I don't know what a bounded topological space is, but it is bounded as a metric space with the usual metric.
    Last edited: Aug 23, 2009
  5. Aug 23, 2009 #4
    Yes, that's what I meant: a closed bounded metric space (closed in itself).
    Thanks for clarifying this for me.
  6. Aug 23, 2009 #5
    boundedness isn't really a topological property, for any metrizable space, one can introduce a new metric:

    d' = min{d, 1}

    and the resulting topology will be the same and the metric will be bounded. More obviously and specifically to your example, (a,b) is homeomorphic to the real line, which isn't bounded. (hence, boundedness isn't really a topological property).
  7. Aug 23, 2009 #6
    Thanks tim_lou
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