---
tag: 0NMGJJ19HD
kind: statement
role: proposition
name: Epsilon-net characterisation of total boundedness
msc: ["54E35"]
uses: [0WE5BBHGM6, 0DC3MPR7C9]
sources:
  - url: https://en.wikipedia.org/wiki/Totally_bounded_space
    locator: In metric spaces
---

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $S \subseteq X$.
Then $S$ is [[totally bounded|0WE5BBHGM6]] if and only if for each real $\varepsilon > 0$ there is a finite set $F \subseteq S$ with
$$S \subseteq \bigcup_{y \in F} B(y, \varepsilon).$$

## Remarks

Such a finite set $F$ is an $\varepsilon$-net of $S$. Here $B(y, \varepsilon)$ is an [[open ball|0DC3MPR7C9]].
Proof: an $\varepsilon$-net gives a cover by balls of diameter at most $2\varepsilon$. Conversely, from a cover by sets $A_k$ of diameter less than $\varepsilon$, choose one point of $S$ in each $A_k$ that meets $S$.
The centres can be taken in $S$ or in $X$. Both choices give the same notion.

## Explanations

### L1 human

For each $\varepsilon$, you can place finitely many points in $S$ such that each point of $S$ is within $\varepsilon$ of one of them.
For $[0, 1]$ and $\varepsilon = 0.1$, the points $0, 0.1, 0.2, \ldots, 1$ do the job.