Explanation of the topology on Rational Numbers

Question

Is there any notes, lecture notes, journals and such that caters "Topology on Rational Numbers Q"? Please help.

Answer 1

As von Eitzen mentions, this material is fairly standard in any introductory topology text. For instance, you can find many of the relevant statements and their proofs on ProofWiki: the rational numbers are a totally separated but non-discrete countable metric subspace of $\Bbb R$, which as a subspace is dense $F_\sigma$ but not $G_\delta$. Other intrinsic facts about its topology (and most of their proofs) are collected on this subpage, although if you run around in the wiki for a while you'll find several more which are not.

ProofWiki is not the most easy site to read, in my opinion, but I don't think you're going to find a more complete collection of results in one place, just because the material is so standard.

Answer 2

When we talk about topology on $\mathbb{Q}$, we usually refer to the induced topology on $\mathbb{Q}$ from the standard topology $\mathbb{R}$ (with the standard metric).