Bounded Set

A set is called bounded if all of its points are within a certain distance of each other. Conversely, a set which is not bounded is called unbounded

For real numbers, a set $S\subset \mathbb{R}$ is bounded if there exists $d>0$ such that for all $x,y\in S$ we have $\left|x-y\right|<d$

This idea could be generalized to metric spaces $(M,d)$, by replace the $\left|x-y\right|$ above with $d(x,y)$