Is a Metric Space with Infinite Distance Totally Bounded?

[johnqwertyful]

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?

 

[johnqwertyful]

http://www.math.ucla.edu/~tao/resource/general/121.1.00s/compact.pdf

in this proof of totally bounded implies bounded, he takes the maximum of d(xi,x1), which he assumes to be finite. So if we assume that the distance between every two points is finite, then we get totally bounded implies bounded.

 

[jgens]

So if we assume that the distance between every two points is finite

This "assumption" is incorporated into the definition of metrics.

 

[shortydeb]

the axioms for a metric space state that for any two points in the metric space, their distance is a real (and finite) number.