Bounded set

A bounded set in a metric space $X$ (with metric $\rho$) is a set $A$ whose diameter $$ \delta(A) = \sup_{x,y \in A} \rho(x,y) $$ is finite.

A bounded set in a topological vector space $E$ (over a field $k$) is a set $B$ which is absorbed by every neighbourhood $U$ of zero (i.e. there exists an $\alpha \in k$ such that $B \subseteq \alpha U$).

A bounded set in a partially ordered set $P$ (with order $\le$) is a set $C$ for which there exist elements $u, v \in P$ such that $u \le x \le v$ for all $x \in C$.

The three definitions coincide in the case of subsets of the real numbers.