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[TL]. So is is not a topology for X since X ¢ Sq and so it does not satisfy [T1]. Sj9 isnot a —, , topology for X since {a,b} € Sy and {b,c} € 819 but {a,b} {b,c} = {b} # 3}9 and so it does, not satisfy [T2]. 3, is not a topology for X since {b}¢%,, and {c}e 3,1 but, {b} Ute} = {b,c} € 3)., , 4.1.1 The Indiscrete Topology, , Let X be any set. Then collection I = {@, X} consisting of the empty set and the whole space is always a, , topology for X called the indiscrete (or trival) topology. The pair (X, I) is called an indiscrete, topological space. [Meerut 2000], , 4.1.2 The Discrete Topology, , *Example 1: Let D be the the collection of all subsets of X, then D is a topology for X called the, discrete topology. The pair (X, D) is called a discrete topological space. [Meerut 2001], , Solution: Since @ < X,X < X, we have @ €D and X eD s0 that [T]] is satisfied. [T2] is, , satisfied. Since the intersection of two subsets of X is again a subset of X. Similarly [T3] is, satisfied since the union of any collection of subsets of X is again a subset of X., , 4.1.3 The Co-finite Topology, , **Example 2: Let X be any set and let 3 be the collection of all those subsets of X whose, complements are finite together with the empty set, that is, a subset A of X belongs to S iff A is empty, or A’ is finite. Show that 3 is a topology for X called the (co-finite topology or the finite, , complement topology)., [Meerut 2007 (B.P.) 2010 (B.P.); Agra 2003, 2006; Bilaspur 2008, 2010; Osmania (A.P.) 2004,, 2007, 2010; Himachal 2003, 2005, 2009; Kanpur 2003; Rohilkhand 2008; Rohtak 2003], , Solution: [T1]: Since X’= © (which is finite) we have X € 3. Also @ € S$ by definition., (T2]:G,G eS = Gj, Gy are finite, = G UG, is finite, => (G, 0G) is finite [De-Morgan Law], 2=GAG «5., (T3]: Ge S¥.A EA = G)is finite ¥A € A, , =arG, rE A} is finite, => [U{G, 2 € A} is finite [By De-Morgan Law], = IG, :AeAteS., , >