Problem 2 (15 points) Let X be a topological space. (i) Show that if a subset A of X is open and if B is a subset of X with A ∩ B = ∅ , then A ∩ B ¯ = ∅ , but A ¯ ∩ B ¯ is not necessarily empty. (ii) Show that (1) A ⊂ B ⇒ int ( A ¯ ) ⊂ int ( B ¯ ) and int ( A ) ¯ ⊂ int ( B ) ¯ (2) A open ⇒ A ⊂ int ( A ¯ ) (3) A closed ⇒ int ( A ) ¯ ⊂ A (iii) Find A ⊂ R such that A , A ¯ , int ( A ) , int ( A ¯ ) , and int ( A ) ¯ are all distinct.