14. A topological space (S, T ) is called connected if S is not the union of two disjoint non-empty open sets.
(a) Prove that if S is connected and f is a continuous function from S onto T , then T is also connected.
(b) Prove that for any a <>b in R, [a, b] is connected. Hint: Suppose [a, b] = U ∪ V for disjoint, non-empty, relatively open sets U and V . Suppose c ∈ U and d ∈ V with c d. Let t := sup(U ∩ [c, d]). Then t ∈ U or t ∈ V gives a contradiction.
(c) If S ⊂ R is connected and c <>d are in S, show that [c, d] ⊂ S. Hint: Suppose c <>t <>d and t ∈/ S. Consider (-∞, t ) ∩ S and (t, ∞) ∩ S.
(d) (Intermediate value theorem) Let a <>b in R and let f be continuous from [a, b] into R. Show that f takes all values between f (a) and f
(b). Hint: Apply parts (a), (b) and (c).