The Cantor Set is making me very confused. I can understand that since only open sets are removed, the Cantor Set if a collection of closed sets. I believe I understand that the Cantor Set has measure zero, and therefore contains only intervals of zero measure. I can see that the endpoints of the segments left behind are never removed, and that there are a (countably) infinite number of them. What I didn't realize until recently is that not every point in the Cantor Set is an endpoint, but it also contains interior points, like 1/4, which is never an endpoint. That makes the Cantor Set uncountable infinite. What confuses me are the endpoints associated with that interior point. Since all intervals in the set must have 0 measure, I think there cannot be an explicit point other than 1/4 to be the boundary of the interval. But if the boundary point cannot be definitively named, in what sense can we call that interval closed? This is where I get all confused. I can only imagine the endpoints associated with 1/4 to be in the neighborhood of 1/4, and so it seems like the definition of an open ball about that point. But it has to be closed, and I can't get a handle on this. Any help in seeing how this works would be greatly appreciated.