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3. Observe that if (X,T) is a discrete space or an indiscrete space, then every open set is a clopen set. Find a topology T on the set X = {a,b,c,d} which is not discrete and is not indiscrete but has the property that every open set is clopen. 4. Let X be an infinite set. If T is a topology on X such that every infinite subset of X is closed, prove that T is the discrete topology. 5. Let X be an infinite set and T a topology on X with the property that the only infinite subset of X which is open is X itself. Is (X,T) necessarily an indiscrete space? 6. () Let T be a topology on a set X such that I consists of precisely four sets; that is, T = {X, 0, A, B}, where A and B are non-empty distinct proper subsets of X. (A is a proper subset of X means that A C X and A + X. This is denoted by A c X.) Prove that A and B must satisfy exactly one of the following conditions: (a) B = X\A; (6) ACB; (c) BCA. [Hint. Firstly show that A and B must satisfy at least one of the conditions and then show that they cannot satisfy more than one of the conditions.] (ii) Using (1) list all topologies on X = {1,2,3,4} which consist of exactly four sets