What is the topological properties of $\mathbb R $ that makes it uncountable (as compared to $\mathbb Q $)?
Further, what axioms (or properties) of $\mathbb R $ do these topological properties depend on? (I suppose completeness, and of course also the ordering, since this is what generates the usual topology...)
There is a proof in Munkres topology that a nonempty compact Hausdorff space which has no isolated points is uncountable. Obviously, this is satisfied by closed intervals in $\mathbb R $, but in $Q $, compact subsets must have isolated points ( this follows from an argument using Baire's category theorem). And can this be shown to follow from completeness?
So can one say that the uncountability of the real numbers hinges on the fact that we have compact sets that are perfect, whereas the set of rationals havn't?
Is it possible to nail down this distinction even further? That is to say that compactness and that closed sets are perfect depend on some other topological property.
Thanks in advance!