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If T is the set $\mathbb{R}$ with the zariski topology then is the set $X=\{0,1\}$ connected?

I think it is connected because the only nonempty subsets are {0,1}, {0}, {1} which are all closed under this topology so therefore there doesn't exist a decomposition into open sets so it is connected. Is this the right reasoning?

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    P.S.: As soon as you find two disjoint, nonempty, complementary closed sets, you've proven your space is not connected; you correctly identified $\{0\}$ and $\{1\}$ as closed sets, but did not notice they were disjoint, nonempty, and complementary. They provide you with a disconnection.2012-03-05

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No, it is not connected. Since $\{0\},\{1\}$ are closed, their complements are open and intersected with $X$ these are just $\{1\},\{0\}$, which are disjoint and cover $X$.