# Separable metric space?

1. May 25, 2007

### pivoxa15

1. The problem statement, all variables and given/known data
'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.

3. The attempt at a solution
Is this correct?

Or they referring to proper subsets only?

2. May 26, 2007

### 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.