First countable spaces and metric spaces.

[alemsalem]

Homework Statement

Show That every metric space is first countable. Hence show that every SUBSET of a metric space is the intersection of a countable family of open sets.

Homework Equations

no equation

The Attempt at a Solution

its easy to show that it is first countable, because for every point in the space there is the set of rational open balls which are included in every other open set.
but the second part of the question is confusing:
how can every subset be an intersection of a countable family? we only know that at every point there is a countable family but. there maybe an uncountable number of point..

Thanks for the help, I've been staring at the question for two hours..
by the way this problem is from P. Szekeres chapter 10 problem 10.9