Prove that if X is a subset of R n and Y is a subset of R m , and X and Y are closed and bounded, then if f : X → Y is continuous and has a inverse function, than the inverse function is also continuous.

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Prove that if   X is a subset of             R        n   and   Y is a subset of             R        m  , and   X and   Y are closed and bounded, then if   f  :  X  →  Y is continuous and has a inverse function, than the inverse function is also continuous.

amanhantmk · Accepted Answer

X  and   Y are - as bounded and closed subsets of             R        n   and             R        m   respectively - compact Hausdorff spaces.In such spaces a set is compact if and only if it is closed. If   f  :  X  →  Y is continuous and   F  ⊆  X is closed then   F is compact so that   f  (  F  )  ⊆  Y is compact as well. The next conclusion is that   f  (  F  ) is closed. So apparantly   f is a closed function, i.e. sends closed sets to closed sets. A continous and closed bijection (i.e. a map that has an inverse) is a homeomorphism. Its inverse will also be a homeomorphism.

Prove that if X is a subset of <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant

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