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

- 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 $A$.

## Subcategories

This category has the following 4 subcategories, out of 4 total.

## Pages in category "Totally Bounded Metric Spaces"

The following 11 pages are in this category, out of 11 total.