Silly Mistake

I was on a math help forum and was trying to help someone with the following problem.

Given two disjoint closed sets and in a metric space , prove that there exists disjoint open sets and such that and .

In my proof I made the following mistake: Let for , . Then .

I should have stopped at this time, since I remember talking about something very similar to this in my analysis class.  It turns out that to gaurantee you need that at least one of to be compact.