I am getting ready for grad school in the fall, and re-teaching myself a bunch of undergraduate subjects. Right now I am reading up on the topology of the real number line. I have come across a fact that is really difficult for me to wrap my head around: The Cantor set is both perfect, and totally disconnected. I am aware that this is the only set (up to homeomorphism) that has both of these properties simultaneously. I guess my question is: Can someone explain to me how it is possible for a set to be both Perfect and totally-disconnected? I am not looking for a proof. I read and understand the proofs, I just can't believe what the logic is telling me. Instead, I am looking for more of a informal explanation. How can a set be closed with no isolation points, but at the same time be totally disconnected!?