Question

Real analysis problem. If C is the cantor middle-third set, then C=C'

How would I prove this. I know that C' ={x: x is a limit point of C}. Also I can prove that C has no intervals, but has unaccountably many elements. Is each of those elements a limit point? Is that all I need to show? @jerico

Answer 1

I'll give you an outline of a proof as I'd do it:

I think that one way to show that two sets are the same is to show that:
1) all elements in \(C'\) are also in \(C\) (\(C'\subseteq C\))
2) all elements in \(C\) are also in \(C'\) (\(C\subseteq C'\))
This means that \(C\) and \(C'\) must be equal.

The Cantor set is topologically closed (and therefore contains all its accumulation points) and every point in the Cantor set is an accumulation point. These two properties should suffice to show that both, conditions 1) and 2), are true.

I propose you try to restate this in more formal terms to see whether it works. I'm just a beginner-level math student so I cannot guarantee that I got this right. But I hope that this helps you to get on the right track.