Definition:Totally Bounded Metric Space

Definition
Let $$(X,d)$$ be a metric space. We say that $$X$$ is totally bounded if for every $$\epsilon > 0$$ there exists a finite family of points $$x_1,\ldots,x_n \in X$$ such that
 * $$X = \bigcup_{i=1}^n B(x_i,\epsilon)$$,

where $$B(x_i,\epsilon)$$ represents the ball of center $$x_i$$ and radius $$\epsilon$$.

This is: $$X$$ is totally bounded if, given any $$\epsilon > 0$$ one can find a finite number of balls of radius $$\epsilon$$ which cover $$X$$.