R2 {(x1, x2) R2\\x1x2 = 1, x2 > 0} Solution Consider the function f: R2 ------>R

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Consider the function    f: R2 ------>R defined as                            f(x,y) = x.y . Note the set { (x,y) | xy = 1} = f-1({1}). That is f is multiplication function and it is a standardexercise to show f is a continuous function. (If you want to knowthe proof ask me. We know that f is continuous function if and only if forevery open set V in R, f-1(V) is open inR2,                                                          if and only if for every closed set F in R, f-1(F)is closed in R2. ----(*) Now consider the set {1} in R. clearly this set is closed inR. Then by (*) f-1({1}) = is closed set inR2 . That is, the given set is closed in R2.Thiscompletes the proof.

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