Why is $\operatorname{spec}(K[x,y])-\{(x,y)\}$ not isomorphic to $\operatorname{spec}K[x,y]$? I cannot see why they can't be an isomorphism between? If both of these are infinite sets there can be a bijection between them. I might be missing some trivial argument, any help is appreciated. I was trying to prove $\{A_k\}^{2}-\{(x,y)\}$ is not affine and I complete the argument if I am able to show the above claim.
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2A bijection is not necessarily a topological isomorphism. – Bernard Mar 19 '18 at 19:48
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1So you're telling me that the plane is topologically isomorphic to the plane minus a point? – Arkady Mar 19 '18 at 19:49
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sorry, but how is it a plane? the plane will correspond to maximal ideals right – user345777 Mar 19 '18 at 19:52
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Related: https://math.stackexchange.com/questions/122821 – Watson Mar 19 '18 at 19:56
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You're right: it's not trivial. See the answer by user1971 here. https://math.stackexchange.com/questions/19884/regular-functions-on-the-plane-minus-a-point/19961 – Billy Mar 19 '18 at 19:56
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And on schemes, even maps which are homeomorphisms at the topological level aren't necessarily scheme isomorphisms. – Daniel Schepler Mar 19 '18 at 19:57
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I just want to show that they are not topologically isomorphic – user345777 Mar 19 '18 at 20:01
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@Billy I don't completely understand how does it solve my problem. – user345777 Mar 19 '18 at 20:08
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See the explanation of Ravi Vakil in the section 4.4.3 of his Math 216 notes. – danneks Mar 20 '18 at 07:06