1. Prove that the intersection of any collection of compact sets is compact 2. (a) Let f be a collection of disjoint open subsets of R. Prove that f is countable. (b) Find an example of a collection...

1. Prove that the intersection of any collection of compact sets is compact

2. (a) Let f be a collection of disjoint open subsets of R. Prove that f is countable.

(b) Find an example of a collection of disjoint closed subsets of R that is not countable.

3. If S is a compact subset of
 and T is a closed subset of S, then T is compact.

(a) Prove this using the definition of compactness.

(b) Prove this using the Heine−Borel theorem.