Consolidate S50 and S51 into S27 (rationals)

spaces/S000027/README.md

@@ -2,15 +2,48 @@
 uid: S000027
 name: Rational numbers
 aliases:
-  - Rationals
+  - $\mathbb Q$
+  - Furstenberg topology
+  - Evenly spaced integer topology
+  - The $p$-adic topology on $\mathbb{Z}$
 counterexamples_id: 30
 refs:
 - doi: 10.1007/978-1-4612-6290-9 
   name: Counterexamples in Topology
 - wikipedia: Rational_number
   name: Rational number on Wikipedia
+- doi: 10.48550/arXiv.1210.1008
+  name: Two topological uniqueness theorems for spaces of real numbers (M. Francis)
+- wikipedia: Furstenberg_topology
+  name: Furstenberg topology on Wikipedia
 ---
-Let $X = \mathbb{Q}$, the set of rational numbers, with the subspace topology induced from the Euclidean real line.
+Let $X=\mathbb Q$, the set of rational numbers, with the subspace topology induced from {S25}.
 
 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, {P53} space without isolated point is homeomorphic to $\mathbb Q$.
+- Every countably infinite, {P5} {P28} space without isolated point is homeomorphic to $\mathbb Q$.
+
+For a proof, see [Countable metric spaces without isolated points (A. Dasgupta)](http://at.yorku.ca/p/a/c/a/25.pdf) 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 {T212} 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 $p,q\in\mathbb Q$.