Solve problem 2.54 and theorem 2.55 in detail please. Please only attempt the questions when you know the right answer please. Definition 2.50. Let X and Y be sets. The product of X and Y, denoted X x Y, is theset of ordered pairs given byX × Y := {(x, y) | x € X ^ y € Y }.Exercise 2.51. Let A, C C X and B, D CY.(1) Show that A × B c X ×Y.(2) Prove or disprove: (X \ A) × (Y \B) = (X ×Y)\(A × B). (If disproved, what doeshold?)(3) Show that (A × B) (C x D) = (AnC) × (Bn D).(4) Can we replace n by U in the above statement? (If not, what does hold?)Definition 2.52. Let (X, Tx) and (Y, Ty) be spaces. The product topology on X × Y is thetopology generated by the basisB := {U × V c X × Y | U e Tx ^V € Ty}.Exercise 2.53. Show that the product topology is well-defined, that is, B in Definition 2.52is a basis for a topology.Problem 2.54. Give an example to show that the basis for the product topology on X × Yis just a basis, and not generally a topology.Theorem 2.55. Let X and Y be spaces, АсХ, ВСҮ, аnd give X xY the producttopology. Then Ax В — А x В.

Name: Solve problem 2.54 and theorem 2.55 in detail please. Please only atte
Uploaded: 2024-03-20T06:38:44.000Z
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Description: Solve problem 2.54 and theorem 2.55 in detail please. Please only attempt the questions when you know the right answer please. Definition 2.50. Let X and Y be sets. The product of X and Y, denoted X x Y, is theset of ordered pairs given byX × Y := {(