Rational numbers

Let $X=\mathbb Q$, the set of rational numbers, with the subspace topology induced from Euclidean Real Numbers.

Defined as counterexample #30 ("The Rational Numbers") in DOI 10.1007/978-1-4612-6290-9.

A purely topological characterization of the space $\mathbb Q$, due to Sierpinski, is given by the following equivalent theorems:

• Every countably infinite, Metrizable space without isolated point is homeomorphic to $\mathbb Q$.
• Every countably infinite, $T_3$ First countable space without isolated point is homeomorphic to $\mathbb Q$.

For a proof, see Countable metric spaces without isolated points (A. Dasgupta) or DOI 10.48550/arXiv.1210.1008. And for the equivalence between the two theorems, the second one easily follows from the first; and the reverse direction follows from (Countable ∧ First countable) ⇒ Second countable together with Urysohn's metrization theorem.

It follows for example that the following spaces are homeomorphic to $\mathbb Q$.

(1) The space $\mathbb{Z}$ with the Furstenberg topology, generated by all sets of the form $a + k \mathbb{Z} = \{a+kn : n\in\mathbb{Z}\}$ with $a\in\mathbb Z$ and positive integer $k$.

Defined as counterexample #58 ("Evenly Spaced Integer Topology") in DOI 10.1007/978-1-4612-6290-9.

(2) The space $\mathbb{Z}$ with the topology generated by all sets of the form $a+p^\alpha\mathbb Z=\{a+p^\alpha n:n\in\mathbb Z\}$ with $a\in\mathbb Z$ and non-negative integer $\alpha$. Here $p$ is a fixed prime.

Defined as counterexample #59 ("The $p$-adic Topology on $\mathbb{Z}$") in DOI 10.1007/978-1-4612-6290-9.

(3) The product space $\mathbb Q \times \mathbb Q$.

(4) The Sorgenfrey topology on $\mathbb Q$, generated by all half-open intervals $[p,q)\cap\mathbb Q$ with $p,q\in\mathbb Q$.