I need some help with a topological proof, hopefully there is someone out there that can help.

Prove that a bijection f: X→Y is a homeomorphism if and only if f and f^-1 map closed sets to closed sets.

Thanks

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- March 25th 2008, 02:21 PMreagan3ncTopology Proof
I need some help with a topological proof, hopefully there is someone out there that can help.

Prove that a bijection f: X→Y is a homeomorphism if and only if f and f^-1 map closed sets to closed sets.

Thanks - March 25th 2008, 03:30 PMiknowone
The definition of a homeomorphism is a map which is bijective continuous and open (maps open sets to open sets).

Assume is a homeomorphism.

is continuous and open so is continuous and open. But then is closed (maps closed sets to closed sets) and so is since let be a closed subset of . Then is open in and so is open in . But and therefore is closed in . Same argument shows is a closed map.

Now assume that and are bijective maps which take closed sets to closed sets. We need to show that is continuous and open. Consider an arbitrary open set . is closed and is closed in . Therefore is open in . So is continuous. Similar argument shows is open. - March 26th 2008, 04:37 AMreagan3nc
THanks, but i have one question are C^c and U^c the compliments of C and U? Just want to make sure.

Thanks - March 26th 2008, 06:58 AMThePerfectHacker