Definition:Totally Bounded Metric Space

From ProofWiki
Jump to navigation Jump to search

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:
$\displaystyle A = \bigcup_{i \mathop = 1}^n \map {B_\epsilon} {x_i}$
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:
$\displaystyle \inf_{0 \mathop \le i \mathop \le n} \map d {x_i, x} \le \epsilon$
for all $x \in A$.


Also known as

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


Also see


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

Internationalization

Totally bounded is translated:

In Dutch: totaal begrensd
In Dutch: precompact