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EN

Art, Music, and Creative Writing homework help

Let f : R n ? R m be continuous, where we equip R n and R m with the Euclidean metric d2. Prove that the following conditions are equivalent: (i) For every bounded set T ? R m, we have that f ?1 (T) is bounded. (ii) For every compact set T ? R m, we have that f ?1 (T) is compact. [Hint: To show that (ii) implies (i) you may have to prove that cl(f ?1 (T)) ? f ?1 (cl(T)).] help pleaseExercise 7 (12 points). Let f: R” —> Rm be continuous, where we equip R” and R“ with theEuclidean metric d2.Prove that the following conditions are equivalent: (i) For every bounded set T C R”, we have that f_1(T) is bounded.(ii) For every compact set T C R”, we have that f ’1(T) is compact. [Hint To Show that (ii) implies (i) you may have to prove that cl( f ’1(T)) C f’1(cl(T)).]

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