Bounded set

Definition

Let $(X, d)$ be a metric space and let $S \subseteq X$. The set $S$ is bounded when there is a real number $M$ such that $d(x, y) \le M$ for all $x, y \in S$.

Remarks

The empty set and each finite set are bounded. If $X$ is not empty, then $S$ is bounded if and only if $S$ lies in some closed ball $\bar{B}(x, r)$ with $x \in X$ and $r \ge 0$. In $\mathbb{R}$ with the usual metric, $S$ is bounded in this sense if and only if $S$ is bounded above and bounded below. Boundedness depends on the metric: $\mathbb{R}$ is not bounded for $|x - y|$, but it is bounded for $\min(|x - y|, 1)$.

Explanations

L1 human

A set is bounded when you can put all of it inside one large ball. For example, the interval $(0, 1)$ is bounded, but the set $\mathbb{N}$ of natural numbers is not bounded in $\mathbb{R}$.