Finding open, disjoint sets Let X be compact and Hausdorff. A, B be closed, disjoint subsets of X. Find U containing A, V containing B (U and V open and disjoint).
My thought would be to fix x in A and pick y in B. By Hausdorff I can find open sets C_xy containing x and D_xy containing y such that C and D are disjoint. If I do this over all y in B I get a collection of open sets D' that cover B. By compactness of X and closedness of B, B is compact. So I can pick a finite subcover of B, D''. The union of this open subcover is my V.
I can then pick the corresponding open sets, C'', about x and intersect them, call it C'''. This set is disjoint from Y. 
Repeat this for all x in A. Union all such C''' and call it U. U and V are open and disjoint.
This sound okay?

You need to take the intersection of your finite subcover to get $V$; that's why the compactness condition is so important here.
Other than that, you have the right idea.
