# If every closed ball in a metric space $X$ is compact, show that $X$ is separable.

The problem says:

If every closed ball in a metric space $X$ is compact, show that $X$ is separable.

I'm trying to use an equivalence in metric spaces that tells us: let X be the matric space, the following are equivalent

X is 2nd countable

X is Lindeloff

X is separable

I also thought about taking balls of a "big" radius and that they are disjoint, but I do not see how to make the set of balls is countable.

• Wouldn't the entire space necessarily be compact as well ('ball of infinite radius')? – Justin Benfield Aug 11 '17 at 18:27
• @JustinBenfield That's what the problem says, it's from a book – 447395 Aug 11 '17 at 18:44

## 2 Answers

For a different approach, choose $x\in X$ so that $X = \cup_{n \in \mathbb{N}} \overline B_x(n).$ Now consider, for fixed $n\in \mathbb N,$ the fact that the compact ball $\overline B_x(n)$ is totally bounded. This means that, for each $m\in \mathbb N$, there is a $\textit{finite}$ set of points $x_{m,n}\in \overline B_x(n)$ such that if $y\in \overline B_x(n)$ then there is an $m\in \mathbb N$ and $x_{m,n}$ such that $d(y,x_{m,n})<1/m,$ which implies that $\left \{ x_{m,n} \right \}_{m\in \mathbb N}$ is a countable dense subset of $\overline B_x(n)$ and therefore that $\left \{ x_{m,n} \right \}_{m,n\in \mathbb N}$ is a countable dense subset of $X$. The result follows by the equivalence separable $\Leftrightarrow$ second countable.

$X = \cup_{n \in \mathbb{N}} D(p, n)$, for any $p \in X$ (where $D(x,r)$ denotes the closed ball around $p$ of radius $r$).

So $X$ is $\sigma$-compact hence Lindelöf hence separable.

• At the end, after “Lindelöf hence separable”, you could have added “since $X$ is a metrice space”. – José Carlos Santos Aug 11 '17 at 18:55
• @JoséCarlosSantos the OP already mentioned it. – Henno Brandsma Aug 11 '17 at 19:08
• You're right. Sorry. – José Carlos Santos Aug 11 '17 at 19:42
• @HennoBrandsma Where have you used that the closed balls are compact? Instead you used that X is compact – 447395 Aug 11 '17 at 19:46
• @Ali I write $X$ as a countable union of closed balls (this is always true) and so a countable union of compact sets. – Henno Brandsma Aug 11 '17 at 19:52