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You are not logged in. #1 20080717 02:01:41
Pasting lemmaLet and be topological spaces and suppose and are subsets of X such that . Let and be continuous functions such that for all . Then defined byis continuous. Proof: Let be Yopen and consider . Then is Aopen and is Bopen and so there exist Xopen sets and such that and . Hence Therefore is Xopen, showing that is continuous. #2 20080717 13:01:07
Re: Pasting lemmaI made a mistake in my proof. I need to find an Xopen set U such that . And I may have to assume that and are closed in as well. Last edited by JaneFairfax (20080717 18:01:23) #3 20080717 18:01:05
Re: Pasting lemmaOkay, I think we can say that if and are either both open or both closed in , then is continuous.If and are both open in , then and are both open in ; hence is open in . If and are both closed in , we’ll use this result: is continuous if and only if given any closed subset , is closed in . So if is closed in , is closed in ; where is closed in . Hence is closed in . Similarly (where is closed in ) is closed in . Thus is closed in . #4 20080717 18:39:57 