Totally bounded

In topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitely many subsets of every 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. For subsets of a complete metric space these meanings coincide but in general they do not. See also use of the axiom of choice below.

A metric space ${\displaystyle (M,d)}$ is totally bounded if and only if for every real number ${\displaystyle \epsilon >0}$, there exists a finite collection of open balls in ${\displaystyle M}$ of radius ${\displaystyle \epsilon }$ whose union contains ${\displaystyle M}$. Equivalently, the metric space ${\displaystyle M}$ is totally bounded if and only if for every ${\displaystyle \epsilon >0}$, there exists a finite cover such that the radius of each element of the cover is at most ${\displaystyle \epsilon }$. This is equivalent to the existence of a finite ε-net.

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