1. Prove or give a counterexample: If a set S has a maximum and a minimum, then S is a closed set. 2. Let A be a nonempty open subset of  and let be the set of rationals. Prove that A ∩  ≠ ∅. 3. Let S...

1. Prove or give a counterexample: If a set S has a maximum and a minimum, then S is a closed set.

2. Let A be a nonempty open subset of
 and let
be the set of rationals. Prove that A ∩
 ≠ ∅.

3. Let S and T be subsets of
. Prove the following.

(a) cl (cl S ) = cl S

(b) cl (S ∪ T ) = (cl S ) ∪ (cl T )

(c) cl (S ∩ T ) ⊆ (cl S ) ∩ (cl T )

(d) Find an example to show that equality need not hold in part (c).