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# sols24 - 56 24 December 1st Topology II 24.1 I have allowed...

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56 24. December 1st: Topology II. 24.1 . I have allowed the endpoints of the open intervals considered in Defini- tion 24.5 to be ± . Prove that it is not necessary to do this, in the sense that one defines the same topology by ‘just’ taking unions of bounded open intervals ( a, b ) with a, b R (and a < b ). Answer: The point here is simply that the extended open intervals ( -∞ , b ), ( a, ), and ( , ) are themselves unions of bounded open intervals: ( a, ) = b>a ( a, b ) ( -∞ , b ) = a<b ( a, b ) ( -∞ , ) = a> 0 ( a, a ) . Thus every ‘union of open intervals’ as in Definition 24.5 can in fact be rewritten as a union of bounded open intervals. 24.2 . Let R stan , resp. R Zar be the topological spaces determined by the standard, resp. Zariski topology on R . Prove that the identity R R is a continuous function R stan R Zar , and that it is not a homeomorphism. Answer: We have to verify that if a set is open in the Zariski topology, then it is open in the standard topology; and that the converse is not necessarily true.

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sols24 - 56 24 December 1st Topology II 24.1 I have allowed...

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