(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'.

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A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…