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I know some examples of spaces which are not totally bounded. For example, the real space $R$ with discrete metric is bounded but not totally bounded. I understand its not totally bounded because the $\epsilon$-nets required to cover the space in this case would be infinite.

Are there other examples of spaces which are not totally bounded but are finite? In my above example, $\mathbb{R}$ is infinite. I was just curious to know if all finite sets are totally bounded.

And also how is total boundedness related to open and closet sets? In a metric space, can an open set be totally bounded? Or the set has to be closed for being totally bounded.

Also, if I have a space $X=(0,1)$ with the absolute value metric, is this space totally bounded?

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  • $\begingroup$ By "finite", do you mean that the space contains only finitely many points? Then every such space is totally bounded. $\endgroup$ – Daniel Fischer Jun 23 '15 at 15:43
  • $\begingroup$ Yes, that's exactly what I mean. It has finite points. In that case, all finite spaces are totally bounded? $\endgroup$ – Khushboo Jun 23 '15 at 15:44
  • $\begingroup$ We have a result that states "If $X=(X,d)$ is a metric space and $A \subseteq X$ then $A$ is totally bounded if and only if closure$(A)$ is totally bounded." $\endgroup$ – Error 404 Jun 23 '15 at 15:50
  • $\begingroup$ Well, if $X = \{x_1, \dotsc, x_n\}$, then forget about the radius and for all $\varepsilon > 0$ the finitely many balls $B_\varepsilon (x_k),\, 1 \leqslant k \leqslant n$, cover $X$. $\endgroup$ – Daniel Fischer Jun 23 '15 at 15:53

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