firstly i thank you for taking interest in my post but i am new here so if i have made some mistakes or done something which is out of place please point out.my problem is we know that the cantor middle onethirds set is uncountable.(i know the proof of that) i am going to prove it is countable,please point out the mistake in my argument here. the cantor middle one thirds set is created by eliminating open intervals from [0,1]. all these intervals are disjoint,hence countable.now the cantor set is made up of the boundary points of these intervals and each interval has two boundary points and the number of intervals are countable.hence the cantor set is countable. this is my argument. i am seriously confused here.please help out.
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closed as too localized by Felipe Voloch, Yemon Choi, Qiaochu Yuan, Andres Caicedo, Simon Thomas Jul 30 '11 at 22:15
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There are points in the Cantor set that are not endpoints of any of the removed intervals. For example $1/4$ is such a point. 

