Fix S72 description (#1092)

Author: StevenClontz

spaces/S000072/README.md

@@ -6,15 +6,21 @@ refs:
   - doi: 10.1007/978-1-4612-6290-9
     name: Counterexamples in Topology
 ---
-Let $S=((0,1)\setminus\{\frac{1}{2}\})\times(0,1)$ and $X=S\cup\{\langle 0,0\rangle,\langle 1,0\rangle\}\cup\{\langle\frac{1}{2},r\sqrt{2}\rangle:r\in(0,\frac{1}{\sqrt 2})\cap\mathbb Q)\}$.
 
-This space is $X$ where $S$ has its subspace topology from {S176}. Then let
+Let $S=\big((0,1)_{\mathbb Q}\setminus\{\frac{1}{2}\}\big)\times(0,1)_{\mathbb Q}$ and
+$X=S\cup\{\langle 0,0\rangle,\langle 1,0\rangle\}\cup\{\langle\frac{1}{2},r\sqrt{2}\rangle:r\in(0,\frac{1}{\sqrt 2})_{\mathbb Q}\}$
+(where $I_{\mathbb Q}=I\cap\mathbb Q$).
 
-- $U_n(0,0) = \{(0,0)\} \cup \{(x,y): 0<x<\frac{1}{4}, 0<y<\frac{1}{n}\}$,
-- $U_n(1,0) = \{(1,0)\} \cup \{(x,y): \frac{3}{4}<x<1, 0<y<\frac{1}{n}\}$, and
-- $U_n(\frac{1}{2},r\sqrt{2}) = \{(x,y): \frac{1}{4}<x<\frac{3}{4}, |y - r\sqrt{2}|<\frac{1}{n}\}$
+This space is $X$ with the topology such that points of $S$ have their usual neighborhoods as a subspace of {S176}, and
 
-be local bases at $(0,0)$, $(1,0)$ and $(\frac{1}{2},r\sqrt{2})$ respectively.
+- $U_n(0,0) = \{\langle 0,0\rangle\} \cup \left((0,\frac{1}{4})_{\mathbb Q}\times(0,\frac{1}{n})_{\mathbb Q}\right)$,
+- $U_n(1,0) = \{\langle 1,0\rangle\} \cup \left((\frac{3}{4},1)_{\mathbb Q}\times(0,\frac{1}{n})_{\mathbb Q}\right)$, and
+- $U_n(\frac{1}{2},r\sqrt{2}) = \left((\frac{1}{4},\frac{3}{4})\times(r\sqrt{2}-\frac{1}{n},r\sqrt{2}+\frac{1}{n})\right)\cap X$
+  for $r\in(0,\frac{1}{\sqrt 2})_{\mathbb Q}$.
 
-Defined as counterexample #80 ("Arens Square")
-in {{doi:10.1007/978-1-4612-6290-9}}.
+are local bases at $\langle 0,0\rangle$, $\langle 1,0\rangle$ and $\langle\frac{1}{2},r\sqrt{2}\rangle$ respectively.
+
+Defined as counterexample #80 ("Arens Square") in {{doi:10.1007/978-1-4612-6290-9}}, where
+this space was erroneously claimed to be {P4}.
+Compare with {S80} which modifies the construction to
+obtain a valid {P4} example.