udg Exercise: If X = {a,b,c,d,e} Determine whether or not each of the following collections of subsets of X is a topology on X (1) r, = {X ,ø.{a},{c,d},{a,c,d}}. (2) r, = {X ,ø.{a},{c,d},{b,c,d}}. (3) 1, = {X ,ø.{a},{a,b},{a,d},{b,e}.{a,b,c}}. (4) 7, = {X ,ø,{a},{c,d},{a,c,d},{b,c,d,e}} (5) r, = {X ,ø.{a,b,e}.{a,b,d},{a,b,d,c}}. %3D %3D .mportant Examples of Topoogies: Definition: Let X be any set. Then T ={X,Ø} is the indiscrete topology on X . Proof: 01) X ,ØeT.

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Q: Let X={a,b,c,d}, then one of the following is a base for some topology on X: O a. {{a,b},{a,c},{a}}…

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Q: Exercise 1. Let X = {a, b, c, d, e}. Which of the following families is a topology? Why or why not?…

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Q: Show that there is no topology on X={a,b,c,d,e} based on the family B={{a,b},{a,b,d},{b,d,e}}

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Q: 4) LetXbe any set with the discrete topology. Let Y be any subset of X. Prove that ô( Y ) =Ø.

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Q: 3. Let X = {a,b,c ,d, e}, f ={{a,b}.{b.c}.fc.e}.fe}}cP(X) Find the topology z on X generated by B.

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Q: Prove the following: Let X denote the set {0,1} with the discrete topology. Let Y = X" = || X; ieZ+…

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