Is every countably infinite subset of a metric space closed?

Question

It seems that the answer is no, but I'm struggling to think of a proper example.

Would I be best trying to think of a closed set whose complement is open?

The complement of a closed set is always open... (that is the definition of a closed set, by the way) — Netchaiev, Feb 08 '18 at 20:21

score 2 · Answer 1 · answered Feb 08 '18 at 20:17

2

$ S:=\left\{\frac{1}{n}: n\ge 1\right\} $ is neither closed nor open (in the usual topology of $\mathbf{R}$).

answered Feb 08 '18 at 20:17

Paolo Leonetti

score 2 · Accepted Answer · answered Feb 08 '18 at 20:19

2

$\mathbb{Q}$ is not closed in $\mathbb{R}$ (with the usual metric)

answered Feb 08 '18 at 20:19

Netchaiev

Didn't realise the answer could be as simple as that. Thanks very much – Dazzler95 Feb 08 '18 at 20:33
You're welcome! – Netchaiev Feb 08 '18 at 20:35

2 Answers2