Metric Space
A subset S of a metric space (M, d) is bounded if it is contained in a ball of finite radius, i.e. if there exists x in M and r > 0 such that for all s in S, we have d(x, s) < r. M is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself.
- Total boundedness implies boundedness. For subsets of Rn the two are equivalent.
- A metric space is compact if and only if it is complete and totally bounded.
- A subset of Euclidean space Rn is compact if and only if it is closed and bounded.
Read more about this topic: Bounded Set