I thought I solved this until I've seen fedja's comment. The subset in question is not very exotic, its just described in an obscure way (like writing "the set of all non trivial solutions for x^3+y^3=z^3" for the empty set)...

Tim: I thought of a solution but perhaps its different from yours because its not a two word phrase. A good way is to enumerate the rationals and for every real number $r$ let $A_r$ be the set of all those $n$ such that $q_n < r$. I thought of two candidate two word phrases: omega-one and "enumerating rationals", but the first doesn't work and the second is very far from a complete solution

Maybe you should change it to "sets for which it is hard to guess their cardinality"? Then there's room for other nice questions like the cardinality of the set of all continuous real valued functions (this example is not difficult, though, but maybe there are others in this vain)

Very nice. How to prove that $f$ must be a polynomial if $F$ is uncountable?

Better: every countable metric (not just compact) can be embedded in Q, and every countable compact metric is homeomorphic to a countable ordinal

Pete - yes, I mean exactly that

What a nice answer!

I mean Apollo's example, not mine:)

In my example the space is not second countable. For every real number $x$ the open interval $U=(x-1,x')=(x-1,x]$ has maximal element $x$. A basis element contained in $U$ and containing $x$ must also have maximal element $x$ so there is a surjection from the set of all basis element to the reals

What is the linear order the Sorgenfrey line is a subspace of? Is this linear order separable, too?