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U xV, where U is an

U xV, where U is an

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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Could you show me how to do 3.35 in detail?

Definition. Let X and Y be two sets. The projection functions Tx : X × Y → X and Ty : X × Y → Y are defined by Tx(x, y) = x and Ty(x, y) = y. Definition. Suppose X and Y are topological spaces. The product topology on the product X x Y is the topology whose basis is all sets of the form U x V, where U is an open set in X and V is an open set in Y. Theorem 3.35. Show that the product topology on X × Y is the same as the topology generated by the subbasis of inverse images of open sets under the projection functions, that is, the subbasis is {nx'(U)| U open in X}U {Ty'(V)| V open in Y}.
Transcribed Image Text:Definition. Let X and Y be two sets. The projection functions Tx : X × Y → X and Ty : X × Y → Y are defined by Tx(x, y) = x and Ty(x, y) = y. Definition. Suppose X and Y are topological spaces. The product topology on the product X x Y is the topology whose basis is all sets of the form U x V, where U is an open set in X and V is an open set in Y. Theorem 3.35. Show that the product topology on X × Y is the same as the topology generated by the subbasis of inverse images of open sets under the projection functions, that is, the subbasis is {nx'(U)| U open in X}U {Ty'(V)| V open in Y}.
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