Thank you for the replies.
But then does this also mean that the intersection of infinitely many nested closed intervals can never be a singleton? (EDIT: in the sense that there will also still be "something left" in this case as well?)
(Please pardon my ignorance if the above is not analogous...
That is all well and good, but as far as I can tell, the Cantor set is a subset of the set of rationals (which is countable) and every infinite subset of a countable set is countable. Where am I going wrong?