Definition:Totally Bounded Metric Space
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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
- Totally Bounded Metric Space is Bounded, but the converse is not true. (The simplest example is a countable set with the standard discrete metric.)
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- Results about totally bounded metric spaces can be found here.
Internationalization
Totally bounded is translated:
In Dutch: | totaal begrensd | |||
In Dutch: | precompact |