# What is the topological properties of $\mathbb R$ that makes it uncountable (as compared to $\mathbb Q$)?

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.

One topological property of $\mathbb{R}$ that makes it uncountable (more precisely, of size at least contiuum) is connectedness. Any Tychonoff (or even functionally Hausdorff) connected at least two point space has size at least continuum. The connectedness comes from completeness, actually the order completeness and no isolated points is enough. Any (at least two point) densely lineary ordered topological space that comes from complete ordering has size at least continuum.