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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!?