Definition:Totally Bounded Metric Space

Definition

Definition 1

A metric space $M = \struct {A, d}$ is totally bounded if and only if:

for every $\epsilon \in \R_{>0}$ there exists a finite $\epsilon$-net for $M$.

That is, $M$ is totally bounded if and only if:

for every $\epsilon \in \R_{>0}$ there exists a finite set of points $x_1, \ldots, x_n \in A$ such that:

$\ds A = \bigcup_{i \mathop = 1}^n \map {B_\epsilon} {x_i}$

where $\map {B_\epsilon} {x_i}$ denotes the open $\epsilon$-ball of $x_i$.

That is: $M$ is totally bounded if and only if, given any $\epsilon \in \R_{>0}$, one can find a finite number of open $\epsilon$-balls which cover $A$.

Definition 2

A metric space $M = \struct {A, d}$ is totally bounded if and only if:

for every $\epsilon \in \R_{>0}$ there exist finitely many points $x_0, \dots, x_n \in A$ such that:

$\ds \inf_{0 \mathop \le i \mathop \le n} \map d {x_i, x} \le \epsilon$

for all $x \in A$.

Also known as

A totally bounded metric space is also referred to as a precompact space.

Also see

• Equivalence of Definitions of Totally Bounded Metric Space

• Totally Bounded Metric Space is Bounded, but the converse is not true. (The simplest example is a countable set with the standard discrete metric.)

This article, or a section of it, needs explaining. In particular: Provide a link to a proof of this statement. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

• Metric Space Compact iff Complete and Totally Bounded.

• Results about totally bounded metric spaces can be found here.

Internationalization

Totally bounded is translated:

In Dutch:	totaal begrensd
In Dutch:	precompact