1) is a regular space, compact set and closed set,
Then there is open sets U and V exist, such that
2) is compact and hausdorff space and is continuous function
prove there is a closed set non empty such that
For part two, define a sequence of sets in the following way: . Since is continuous and takes a compact space to a Hausdorff space, the closed map lemma applies; we can say that each is a closed set. Let . Note that is closed and . To complete the proof, it necessary to show that is nonempty. I can't see how to do that at the moment, so hopefully somebody else can fill in the gap.
I hope that this gives you some ideas, at any rate.