# Totally bounded but not bounded

1. Mar 3, 2014

### 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?

2. Mar 3, 2014

3. Mar 3, 2014

### jgens

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

4. Mar 4, 2014

### 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.