Let (X, ) and (Y, ) be two topological spaces. According to Result 16.23(i), if is

Chapter 16, Problem 16.28

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Let (X, ) and (Y, ) be two topological spaces. According to Result 16.23(i), if is thediscrete topology on X, then every function f : X Y is continuous. The converse ofResult 16.23(i) is stated as follows together with a proof.Converse of Result 16.23(i): Let (X, ) and (Y, ) be two topological spaces. If everyfunction from X to Y is continuous, then is the discrete topology on X.Proof. Suppose that every function f : X Y is continuous and assume, to thecontrary, that is not the discrete topology on X. Then there exists some subset S of Xsuch that S is not open in X. So S is distinct from X and . Let T be an open set in Yand let a, b Y such that a T and b / T. Define a function f : X Y byf(x) = a if x Sb if x / S.Since T is open in Y and f 1(T) = S is not open in X, it follows that f is not continuous,which is a contradiction.(a) Is the proposed proof of the converse correct?(b) If the answer to (a) is yes, then state Result 16.23(i) and its converse using if andonly if. If the answer to (a) is no, then revise the hypothesis of the converse so thatit is true (with attached proof).