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sols26 - 60 26. December 8th: Topology IV. 26.1. Prove that...

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60 26. December 8th: Topology IV. 26.1 . Prove that R with the Zariski topology is connected. Answer: Let S R be any set other than and R .T h e n S and S cannot both be fnite, since otherwise R itselF would be fnite. It Follows that S cannot both be closed and open in the Zariski topology. So there is no set as required by the defnition oF disconnected. It Follows that R is connected. ° 26.2 . Let f : T 1 T 2 be a surjective continuous Function between topological spaces. Prove that iF T 1 is connected, then T 2 is connected. Answer: We can prove the equivalent (contrapositive) statement: iF T 2 is discon- nected, then T 1 must also be disconnected. ±or this, let S T 2 be a set that is both closed and open, and such that S ° = , S ° = T 2 .S in c e f is surjective, it Follows that f 1 ( S ) ° = , f 1 ( S ) ° = T 1 . Also, f 1 ( T 2 ° S )= T 1 ° f 1 ( S ). (Make sure you are able to veriFy all these claims!) Since S is open and f is continuous, f 1 ( S )is open. Since
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sols26 - 60 26. December 8th: Topology IV. 26.1. Prove that...

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