Totally bounded set

Definition

Let $(X, d)$ be a metric space and let $S \subseteq X$. The set $S$ is totally bounded when for each real $\varepsilon > 0$ there are finitely many sets $A_1, \ldots, A_n \subseteq X$ with $S \subseteq A_1 \cup \cdots \cup A_n$ and $\operatorname{diam}(A_k) < \varepsilon$ for each $k$. The space $X$ is totally bounded when $X$ itself is a totally bounded set.

Remarks

Here $\operatorname{diam}$ is the diameter. Each totally bounded set is bounded. The converse is false: an infinite set with the discrete metric is bounded but not totally bounded. In $\mathbb{R}^n$ with the Euclidean metric, a set is totally bounded if and only if it is bounded. The epsilon-net characterisation gives an equivalent form with balls.

Explanations

L1 human

A totally bounded set can be cut into finitely many small pieces, for each size of piece that you choose. A bounded set only fits in one big ball, and this is not enough in infinite dimensions.