(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let U and V be open sets in R^{n}and let f be a one-to-one mapping from U onto V (so that there is an inverse mapping f^{-1}). Suppose that f and f^{-1}are both continuous. Show that for any set S whose closure is contained in U we have f(bd(S)) = bd(f(S)).

2. Relevant equations

Open sets: every point in the set is an interior point. int(S) = S. Or S contains none of its boundry points.

bd(S) = {x in R^{n}| B(r,x)∩S≠ø and B(r,x)∩S^{c}≠ø for every r>0}

3. The attempt at a solution

ie, show that the function takes a boundry to a boundry.

Let S be a set in U. The let f(S) = T. Note that f^{-1}(T) = S.

Also, since T = {x in R^{n}s.t. f(x) V} and f conts, and V open, then T is open.

The same conclusion holds for S.

Thus T and S are both open. They also have the same number of elements.

I don't know where to go from here.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Conts fncs, open sets, boundry

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