Show that every closed subset of a metric space is the intersection of a countable number of open sets.
This is Armstrong's Basic Topology Exercise 2.30 and it's following the section on Tietze extension theorem. I'm thinking about using the Tietze extension theorem, which states:
Any real-valued continuous function defined on a closed subset of a metric space can be extended over the whole space.
But I have no clue how to construct the open sets that we need to intersect...so maybe I'm not on the right track. Any hint please (to the question itself, not necessarily using Tietze extension theorem?)