Everyone knows the Heine-Borel theorem characterizing compact subsets of Euclidean space. For any $n \in \mathbb N$ a set $A \subseteq \mathbb R^n$ is compact just in case it is closed and bounded (in the sense that it is contained in some large enough finite diameter ball).


There is a similar theorem for the irrationals, due to Rothberger. The irrationals are homeomorphic to the space $\mathbb N^\mathbb N$ of functions $f:\mathbb N \to \mathbb N$ topologized by the product topology of the discrete topology on $\mathbb N$. With this presentation a set $A \subseteq\mathbb N^\mathbb N$ is compact just in case it is closed and bounded where bounded means there is an $f:\mathbb N \to \mathbb N$ so that for all $g \in A$ we have $\forall n \, g(n) \leq f(n)$.


My question is whether there is a similar (or at least relatively neat) characterization of compact sets of rationals (with the subspace topology). Does some relatively simple (topological, geometric, combinatorial) property characterize when a closed $A \subseteq \mathbb Q$ is compact?



Structure theorems for compact sets of rationals
It is easy to describe these up to homeomorphism:


* Every countable compact Hausdorff space is homeomorphic to a countable successor ordinal, see [Milliet - A remark on Cantor derivative](https://arxiv.org/abs/1104.0287v1).
* Conversely, it's easy to embed all countable limit ordinals in the rationals by transfinite induction, and successor ordinals will give you compact sets.


