Although the concept of boundedness is quite understandable, I am struggling with the concept of total boundedness.
If I understand properly, then the metric space $[0,1] \subset \mathbb{R}$ with the ordinary distance $d(x,y)=|x-y|$ is not totally bounded. This because I cannot find a finite union of balls of radius $\varepsilon$, with $\varepsilon$ that can be arbitrary small, that "covers" $[0,1]$.
For example, for $\varepsilon = 0.5$ I can find a finite number of balls to cover $ [0,1]$ (for instance, two balls are enough), for $\varepsilon = 0.2$ I need e.g. five balls, etc. But for arbitrary small $\varepsilon$ I would need an infinite number of balls, and therefore the defined metric space is not totally bounded. Is that reasoning correct?
If so, I I have the feeling that a necessary condition to have totally boundedness is that the set in the metric space must be finite (the condition $\forall \varepsilon>0$ bothers me a lot 🙂 )
Best Answer
The condition is that for every $\epsilon > 0$ there is some finite $N(\epsilon)$ (a natural number) so that $N(\epsilon)$ many balls of radius $\epsilon$ cover $[0,1]$. There is no other condition except that $N(\epsilon)$ is finite for every $\epsilon > 0$, and this number will often grow with $\epsilon$ getting smaller, but that's OK. The finite number depends on $\epsilon$ and that's fine.