I have a question, we can consider it for metric spaces specifically. I wanted to ask if is there any known property a metric space $X$ may posses such that any closed bounded set will imply it is also totally bounded. Also, is there a special name for such a space in the literature? It is true for totally bounded metric spaces, as been pointed out. But does it mean that all metric spaces that fulfill this requirement are totally bounded spaces?
Asked
Active
Viewed 389 times
1 Answers
1
Yes, those are the totally bounded metric spaces. For example it is easy to see that every compact metric space is totally bounded.
S -
- 3,611
- 2
- 18
- 38
-
this is not enough, $\mathbb{R}$ is not totally bounded but it has this property. – User666x Dec 08 '15 at 11:03
-
@User666x It doesn't. It is not possible to cover $\mathbb{R}$ with a finite union of balls of radius $\varepsilon$. – S - Dec 08 '15 at 11:05
-
Yes it does. The property in question is "any closed bounded set is totally bounded". That's true of $\Bbb R$. – David C. Ullrich Dec 08 '15 at 14:19