4. Suppose that E ~ X and f : E -t Y.

(a) Let E ~ X and f : E -t Y. Prove that f is continuous on E if and only if f-l(A) n E is relatively closed in E for all closed sets A in Y.

(b) Suppose that f is continuous on E. Prove that if V is relatively open in f(E), then f-l(V) is relatively open in E, and if A is relatively closed in f(E), then f-l(A) is relatively closed in E.