Category:Totally Bounded Metric Spaces

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This category contains results about Totally Bounded Metric Spaces in the context of Metric Spaces.
Definitions specific to this category can be found in Definitions/Totally Bounded Metric Spaces.


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 $A$.