Totally bounded space

In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space).

The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general.

In metric spaces

A metric space

(

M

,

d

)

{\displaystyle (M,d)}

is totally bounded if and only if for every real number

ε

>

0

{\displaystyle \varepsilon >0}

, there exists a finite collection of open balls of radius

ε

{\displaystyle \varepsilon }

whose centers lie in M and whose union contains M. Equivalently, the metric space M is totally bounded if and only if for every

ε

>

0

{\displaystyle \varepsilon >0}

, there exists a finite cover such that the radius of each element of the cover is at most

ε

{\displaystyle \varepsilon }

. This is equivalent to the existence of a finite ε-net. A metric space is totally bounded iff every sequence admits a Cauchy subsequence; in complete metric spaces, a set is compact if and only if it is closed and totally bounded.

Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded). The reverse is true for subsets of Euclidean space (with the subspace topology), but not in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded: every discrete ball of radius

ε

=

1

/

2

{\displaystyle \varepsilon =1/2}

or less is a singleton, and no finite union of singletons can cover an infinite set.