1. For i = 1,2 let fi: Xi --> Yi be maps between topological spaces. Show that the product f1Xf2: X1XX2 --> Y1XY2 defined by f1Xf2(x1x2):= (f1(x1), f2(x2)) is continuous if and only if f1 and f2 are continuous.
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Since is continuous, then for any and any neighborhood . We can find a neighborhood , such that . Similarly, since , for any and any ...
This is a proof regarding continuity and products.