Totally Bounded Space - Definition For A Metric Space

Definition For A Metric Space

A metric space is totally bounded if and only if for every real number, there exists a finite collection of open balls in of radius whose union contains . Equivalently, the metric space is totally bounded if and only if for every, there exists a finite cover such that the radius of each element of the cover is at most . This is equivalent to the existence of a finite ε-net.

Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded), but the converse is not true in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded.

If M is Euclidean space and d is the Euclidean distance, then a subset (with the subspace topology) is totally bounded if and only if it is bounded.

Read more about this topic:  Totally Bounded Space

Famous quotes containing the words space and/or definition:

    When my body leaves me
    I’m lonesome for it.
    but body
    goes away to I don’t know where
    and it’s lonesome to drift
    above the space it
    fills when it’s here.
    Denise Levertov (b. 1923)

    No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.
    Florence Nightingale (1820–1910)