Do the following: a) Prove Theorem 8.4, item 3, on continuity of a function. b) State 2 other general characterizations of continuous functions. c) Suppose (X, T) is a topological space where X is a finite set with more than 1 million elements in it and the co-finite topology on X. Prove that every function f (X,T) → (Y,T') is continuous. d) Explain and illustrate the meaning of a homeomorphism. e) Explain and illustrate the meaning of a topological property. Theorem 8.4 Suppose f: (X, T) (Y,T'). - 1) Let B = ran(f) ≤Y. Then f is continuous iff f : (X,T) (B, Tg) is continuous. In other words, B = the range of f (a subspace of the codomain Y) is what matters for the continuity of f; points of Y not in B (if any) are irrelevant. For example, the function sin: RR is continuous iff the function sin: R-[-1, 1] is continuous.