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Definition 2.52. Let (X,Tx) and (Y, Ty) be spaces. The product topology on 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.

Definition 2.52. Let (X,Tx) and (Y, Ty) be spaces. The product topology on 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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve Exercise 2.53 in detail please

Definition 2.50. Let X and Y be sets. The product of X and Y, denoted X x Y, is the set of ordered pairs given by X x Y := {(x, y) | x E X ^ Y E Y}. Exercise 2.51. Let A, C c X and B, DCY. (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 (Ах В)n (С х D) — (AnC)x (BND). (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 B := {U × V C X × Y | U € Tx AVE TY}. Exercise 2.53. Show that the product topology is well-defined, that is, B in Definition 2.52 is a basis for a topology.
Transcribed Image Text:Definition 2.50. Let X and Y be sets. The product of X and Y, denoted X x Y, is the set of ordered pairs given by X x Y := {(x, y) | x E X ^ Y E Y}. Exercise 2.51. Let A, C c X and B, DCY. (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 (Ах В)n (С х D) — (AnC)x (BND). (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 B := {U × V C X × Y | U € Tx AVE TY}. Exercise 2.53. Show that the product topology is well-defined, that is, B in Definition 2.52 is a basis for a topology.
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