I read that the Cantor set is an uncountable perfect set that does not contain isolated points. At the same time, it is totally disconnected and does not contain intervals of any kind.
I could not figure out where the irrational numbers can be located inside the Cantor set. We know that the endpoints of the Cantor set are countable since they are rational numbers. Now, the irrationals should be 'inside' the interval between the rational endpoints…. But if there are no intervals, where would the irrationals lie? s.t. the Cantor ser is classified as uncountable.
Best Answer
The Cantor set has lots of points that are not end points of the intervals that are removed in its construction. The irrational numbers within the Cantor sets are not the end points of those intervals. Your confusion is caused by thinking that every points between $0$ and $1$ is either in one of the intervals removed or an end point of one of those intervals.