## Totally Bounded Space

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.

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### Some articles on totally bounded space:

**Totally Bounded Space**- Use of The Axiom of Choice

... the proof does not require choice) that every precompact

**space**is

**totally bounded**in other words, if the completion of a

**space**is compact, then that

**space**is

**totally bounded**... But it is no longer true (that is, the proof requires choice) that every

**totally bounded space**is precompact in other words, the completion of a

**totally bounded space**might not be ...

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

“As photographs give people an imaginary possession of a past that is unreal, they also help people to take possession of *space* in which they are insecure.”

—Susan Sontag (b. 1933)

“On our streets it is the sight of a *totally* unknown face or figure which arrests the attention, rather than, as in big cities, the strangeness of occasionally seeing someone you know.”

—For the State of Vermont, U.S. public relief program (1935-1943)

“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)