When we use the nested interval property, we have closed intervals that are nested inside of each other and eventually when we do this a countable number of times out to infinity we eventually enclose a singlet point. So my question is can we apply this to the cantor set? I mean when we remove the middle third of our line segment we keep the first third and the last third and they are closed intervals. And if I look at part of the cantor set I will have nested closed intervals that are subsets of each other. But we know that the cantor set has no isolated points. So what is wrong with my reasoning?