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

Kyle Sutton

Kyle Sutton

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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.

Answer & Explanation



Beginner2022-07-03Added 17 answers

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.

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