Please give an example of a function f, and two sets $\displaystyle U_1,U_2:U_1 \subset U_2, U_1 \neq U_2$ but $\displaystyle f^{-1} (U_1)=f^{-1}(U_2)$? Is there such a function in $\displaystyle R^n$?
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Define f to map all of $\displaystyle R^n$ to a single point. The inverse image of any set containing that point is all of $\displaystyle R^n$.
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