Bounded Set - Metric Space

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

Famous quotes containing the word space:

    Play is a major avenue for learning to manage anxiety. It gives the child a safe space where she can experiment at will, suspending the rules and constraints of physical and social reality. In play, the child becomes master rather than subject.... Play allows the child to transcend passivity and to become the active doer of what happens around her.
    Alicia F. Lieberman (20th century)