Let, Q is set of rational numbers with relative topology of natural topology on R. Is it possible to represent Q as the countable union of disjoint dense subsets?
P.S. According the famous book, "Counterexamples in topology" by Steen and Seebach, it is possible, (page 143. Definition of Roys Lattice space). But, I dont know the proof of this.
Sure, there are various ways to do this.
In my opinion, the nicest is to consider restrictions on denominators - for example, let $B_k$ be the set of rational numbers whose lowest-terms representation has denominator a power of $k$ (with the convention that the lowest-terms representation of $0$ is $0\over 1$). Then:
What can you say about each $B_k$ in terms of topology?
How do the different $B_k$s relate? In particular, can you find an infinite set $X\subseteq\mathbb{N}$ such that for distinct $a,b\in X$ we have $B_a\cap B_b=\emptyset$ (which will then give you the partition you want)? HINT: think about divisibility ...