Totally bounded but not bounded

  • #1
395
14
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?
 

Answers and Replies

  • #3
jgens
Gold Member
1,583
50
So if we assume that the distance between every two points is finite

This "assumption" is incorporated into the definition of metrics.
 
  • #4
29
1
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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