connected. It's often impossible (in full generality) to complete arguments such as this by looking at openess/closedness (although it works sometimes, like proving things like the fundamental lemma of path connectedness in alg. top.).
For the argument I'll try to give you two well placed hints:
Hint 1: If then .
Hint 2: If for every finite one has that then for every finite one has .
So, since the are closed we know that the are open and so the above shows we've found a ____________ with no _____________.