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Exercise 2.60. Let X and Y be spaces and give X × Y the product topology. Show that n and T2 have the properties: (1) п] аnd п2 are onto. (2) VU орen in X, п (U) is open in X x Y. (3) VV open орen in Y, n (V) is open in X x Y.

Exercise 2.60. Let X and Y be spaces and give X × Y the product topology. Show that n and T2 have the properties: (1) п] аnd п2 are onto. (2) VU орen in X, п (U) is open in X x Y. (3) VV open орen in Y, n (V) is open in X x Y.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please solve 2.60 all the parts in detail 

2.6. Products of Sets and the Product Topology. Definition 2.50. Let X and Y be sets. The product of X and Y, denoted X × Y, is the set of ordered pairs given by X x Y := {(x, y) | x € X ^ y E Y}. Еxercise 2.51. Let A, C c Х аnd B, D c Y. (1) Show that A × B c X × Y. (2) Prove or disprove: (X \ A) × (Y\B) = (X × Y) \(A × B). (If disproved, what does hold?) (3) Show that (A × B) n (C × 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 x Y is the topology generated by the basis = {U x V c X × Y |U € Tx ^V E Ty}. B:= Exercise 2.53. Show that the product topology is well-defined, that is, B in Definition 2.52 is a basis for a topology. Problem 2.54. Give an example to show that the basis for the product topology on X x Y is just a basis, and not generally a topology. Theorem 2.55. Let X and Y be spaces, A c X, B C Y, and give X x Y the product topology. Then A × B = A × B. Problem 2.56. Let X and Y be spaces, Ас х, В с Ү, аnd give х xҮ the product topology. What holds between А'x В' аnd (Ax В)'? Theorem 2.57. Let X and Y be spaces, Ac X, B C Y, and give X x Y the product topology. Give A and B the subspace topologies. Show that the product topology on A x B is the same as the subspace topology on A × B as a subspace of X × Y . e Theorem 2.63 Theorem 2.58. Let X andY be spaces. Consider the collection S of all subsets of X × Y of the forms: U x Y where U is open in X, and X x V where V is open in Y. Show that S is a subbasis for the product topology on X × Y. Definition 2.59. Let X and Y be sets. The projection functions are the functions T1 : X x Y → X and 12 : X × Y → Y defined by T1((x, y)) = x, respectively, 72((x, y)) = y. Exercise 2.60. Let X and Y be spaces and give X × Y the product topology. Show that T1 and T2 have the properties: (1) п апd Пу are onto. (2) VU орen in X, п "(U) is open in X хҮ. (3) VV орen in Y, п, (V) is open in X x Y.
Transcribed Image Text:2.6. Products of Sets and the Product Topology. Definition 2.50. Let X and Y be sets. The product of X and Y, denoted X × Y, is the set of ordered pairs given by X x Y := {(x, y) | x € X ^ y E Y}. Еxercise 2.51. Let A, C c Х аnd B, D c Y. (1) Show that A × B c X × Y. (2) Prove or disprove: (X \ A) × (Y\B) = (X × Y) \(A × B). (If disproved, what does hold?) (3) Show that (A × B) n (C × 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 x Y is the topology generated by the basis = {U x V c X × Y |U € Tx ^V E Ty}. B:= Exercise 2.53. Show that the product topology is well-defined, that is, B in Definition 2.52 is a basis for a topology. Problem 2.54. Give an example to show that the basis for the product topology on X x Y is just a basis, and not generally a topology. Theorem 2.55. Let X and Y be spaces, A c X, B C Y, and give X x Y the product topology. Then A × B = A × B. Problem 2.56. Let X and Y be spaces, Ас х, В с Ү, аnd give х xҮ the product topology. What holds between А'x В' аnd (Ax В)'? Theorem 2.57. Let X and Y be spaces, Ac X, B C Y, and give X x Y the product topology. Give A and B the subspace topologies. Show that the product topology on A x B is the same as the subspace topology on A × B as a subspace of X × Y . e Theorem 2.63 Theorem 2.58. Let X andY be spaces. Consider the collection S of all subsets of X × Y of the forms: U x Y where U is open in X, and X x V where V is open in Y. Show that S is a subbasis for the product topology on X × Y. Definition 2.59. Let X and Y be sets. The projection functions are the functions T1 : X x Y → X and 12 : X × Y → Y defined by T1((x, y)) = x, respectively, 72((x, y)) = y. Exercise 2.60. Let X and Y be spaces and give X × Y the product topology. Show that T1 and T2 have the properties: (1) п апd Пу are onto. (2) VU орen in X, п "(U) is open in X хҮ. (3) VV орen in Y, п, (V) is open in X x Y.
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