In topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitely many subsets of any fixed "size" (where the meaning of "size" depends on the given context). The smaller the size fixed, the more subsets may be needed, but any specific size should require only finitely many subsets. A related notion is a totally bounded set, in which only a subset of the space needs to be covered. Every subset of a totally bounded space is a totally bounded set; but even if a space is not totally bounded, some of its subsets still will be.
The term precompact (or pre-compact) is sometimes used with the same meaning, but `pre-compact' is also used to mean relatively compact. In a complete metric space these meanings coincide but in general they do not. See also use of the axiom of choice below.
Read more about Totally Bounded Space: Definition For A Metric Space, Definitions in Other Contexts, Examples and Nonexamples, Relationships With Compactness and Completeness, Use of The Axiom of Choice
Famous quotes containing the words totally, bounded and/or space:
“... we performers are monsters. We are a totally different, far-out race of people. I totally and completely admit, with no qualms at all, my egomania, my selfishness, coupled with a really magnificent voice.”
—Leontyne Price (b. 1927)
“Me, what’s that after all? An arbitrary limitation of being bounded by the people before and after and on either side. Where they leave off, I begin, and vice versa.”
—Russell Hoban (b. 1925)
“It is the space inside that gives the drum its sound.”
—Hawaiian saying no. 1189, ‘lelo No’Eau, collected, translated, and annotated by Mary Kawena Pukui, Bishop Museum Press, Hawaii (1983)