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

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### Famous quotes containing the words space, totally and/or bounded:

“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)

“I introduced her to Elena, and in that life-quickening atmosphere of a big railway station where everything is something trembling on the brink of something else, thus to be clutched and cherished, the exchange of a few words was enough to enable two *totally* dissimilar women to start calling each other by their pet names the very next time they met.”

—Vladimir Nabokov (1899–1977)

“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)