(a) Let X and Y be topological spaces, and let X xY be the corresponding product space. Define the projections px : X × Y → X and py : X × Y → Y by px(x, y) = x and py(x, y) = y. Prove that px and py are continuous. (b) Show that the product topology is the coarsest topology for which both functions px and py are continuous. That is, show that if T is the product topology on X × Y and T' is a topology on X x Y such that px and py are continuous, then T CT'.

(a) Let X and Y be topological spaces, and let X xY be the corresponding product space. Define the projections px : X × Y → X and py : X × Y → Y by px(x, y) = x and py(x, y) = y. Prove that px and py are continuous. (b) Show that the product topology is the coarsest topology for which both functions px and py are continuous. That is, show that if T is the product topology on X × Y and T' is a topology on X x Y such that px and py are continuous, then T CT'.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.4: Spanning Sets And Linear Independence

Problem 76E: Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are...

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