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 definition and/or space:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The limerick packs laughs anatomical
Into space that is quite economical,
But the good ones I’ve seen
So seldom are clean
And the clean ones so seldom are comical.”
—Anonymous.