Is every countable metric space separable?

[pivoxa15]

Homework Statement

'In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space.'

http://en.wikipedia.org/wiki/Separable_metric_space

Let (X,d) be a metric space. If X is countable than it immediately satisfies being a separable metric space? Because just choose X itself as the subset. The closure of X must be X. Hence there exists a countable dense subset, namely X itself.

The Attempt at a Solution

Is this correct?

Or they referring to proper subsets only?

 

[AKG]

If they meant proper, they'd say it. Admitting only proper subsets is equivalent to excluding countable sets. There's no good reason for doing that. Also, that article gives an example of a countable space being separable.