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Totally bounded but not bounded

  1. Mar 3, 2014 #1
    It seems strange, but would a metric space consisting of two points, X={a,∞} be totally bounded, but not bounded? because d(a,∞)=∞. But for all ε>0, X=B(ε,a)UB(ε,∞).

    It's been proven that totally bounded→bounded, so this is wrong. Why?
  2. jcsd
  3. Mar 3, 2014 #2
  4. Mar 3, 2014 #3


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    Gold Member

    This "assumption" is incorporated into the definition of metrics.
  5. Mar 4, 2014 #4
    the axioms for a metric space state that for any two points in the metric space, their distance is a real (and finite) number.
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