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I tried to prove that the rational sequence topology is zero dimensional that is X is T1 space and for every x in X , and every closed set C doesnt contain x , there exists a clopen subset U such that U contains x and U and C are disjoint . It is not difficult to prove that R is T1 and also it is not difficult to prove the second condition if we take x in Q as every set in Q is clopen , but l couldnt prove the second condition if x is not in Q , can someone give me a hint?

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All you have to do is show that it has a base of clopen sets. It’s not hard to show that every set in the base described here is clopen.

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