Bounded set

The word "bounded" makes no sense in a general topological space without a corresponding metric.

Therefore, a set of real numbers is bounded if it is contained in a finite interval.

In topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness.

If the topology of the topological vector space is induced by a metric which is homogeneous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions coincide.

However, S may be bounded as subset of Rn with the lexicographical order, but not with respect to the Euclidean distance.