Completing S64 (pointed rational ext. of plane) (#1412)

pzjp · yhx-12243 · prabau · web-flow · commit 1ba4ae0dc522 · 2025-09-08T13:55:09.000+02:00
Co-authored-by: yhx-12243 &lt;yhx12243@gmail.com&gt;
Co-authored-by: Patrick Rabau &lt;70125716+prabau@users.noreply.github.com&gt;
diff --git a/spaces/S000064/README.md b/spaces/S000064/README.md
@@ -1,12 +1,20 @@
 ---
 uid: S000064
-name: Pointed rational extension of the plane
+name: Rational extension of the plane
 counterexamples_id: 72
 refs:
-  - doi: 10.1007/978-1-4612-6290-9 
+  - zb: "0386.54001"
     name: Counterexamples in Topology
 ---
-If $(X,\tau)$ is the Euclidean plane, we define a new topology for $X$ by declaring open each point in the set $D = \{(x,y) | x \in Q, y \in Q\}$, and each set of the form $\{x\} \cup (D \cap U)$ where $x \in U \in \tau$.
+If $(X,\tau)$ is the {S176},
+we define a new topology for $X$ by declaring open each point in the set
+$D = \mathbb Q^2$, and each set of the form $\{x\} \cup (D \cap U)$
+where $x \in U \in \tau$.
+
+The topology is finer than that of {S176}.
+
+The construction combines the ideas from
+{S60} and {S62}.
 
 Defined as counterexample #72 ("Rational Extension in the Plane")
-in {{doi:10.1007/978-1-4612-6290-9}}.
+in {{zb:0386.54001}}.
diff --git a/spaces/S000064/properties/P000010.md b/spaces/S000064/properties/P000010.md
@@ -0,0 +1,10 @@
+---
+space: S000064
+property: P000010
+value: false
+---
+
+Let $p\in X\setminus D$. Take $U$ be an open neighbourhood of $p$ such that $U\setminus D=\{p\}$ (any base for the topology has to contain such an open set).
+By definition, there exists a Euclidean-open $V$ such that $p\in V$ and $V\cap D\subset U$.
+Then $V\subset \mathrm{int}(\overline{V\cap D})\subset \mathrm{int}(\overline{U})$ and $V$ is not contained in $U$.
+This shows that $U$ is not a regular open set.
diff --git a/spaces/S000064/properties/P000011.md b/spaces/S000064/properties/P000011.md
diff --git a/spaces/S000064/properties/P000018.md b/spaces/S000064/properties/P000018.md
diff --git a/spaces/S000064/properties/P000022.md b/spaces/S000064/properties/P000022.md
@@ -3,11 +3,11 @@ space: S000064
 property: P000022
 value: false
 refs:
-- doi: 10.1007/978-1-4612-6290-9_6
+- zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
 The function $f : X \to \mathbb{R}$ defined by $f(x,y) = x$ can be shown to be continuous, and is unbounded.
 
 Asserted in the General Reference Chart for space #72 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+{{zb:0386.54001}}.
diff --git a/spaces/S000064/properties/P000026.md b/spaces/S000064/properties/P000026.md
@@ -3,10 +3,10 @@ space: S000064
 property: P000026
 value: true
 refs:
-- doi: 10.1007/978-1-4612-6290-9_6
+- zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
 $D$ is countable and dense.
 
-See item #3 for space #72 in {{doi:10.1007/978-1-4612-6290-9_6}}.
+See item #3 for space #72 in {{zb:0386.54001}}.
diff --git a/spaces/S000064/properties/P000028.md b/spaces/S000064/properties/P000028.md
diff --git a/spaces/S000064/properties/P000032.md b/spaces/S000064/properties/P000032.md
@@ -3,9 +3,13 @@ space: S000064
 property: P000032
 value: false
 refs:
-- doi: 10.1007/978-1-4612-6290-9_6
+- zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
-Asserted in the General Reference Chart for space #72 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+Consider an open cover consisting of $X\setminus (\mathbb Q\times\{\sqrt 2\})$ and $\{(q,\sqrt 2)\}\cup D$ for $q\in\mathbb Q$.
+Suppose there is a refinement $\mathscr V$ of the above cover. For each $q\in\mathbb Q$ there exists Euclidean-open $V_q$ such that $(q,\sqrt 2)\in V_q$ and
+$V_q\cap D$ is contained in an element of $\mathscr V$.
+Picking a convergent sequence of rationals $q_n \to q$ we
+observe that infinitely many $V_{q_n}$'s have to intersect any neighbourhood of $(q,\sqrt{2})$, therefore $\mathscr V$
+is not locally finite.
diff --git a/spaces/S000064/properties/P000033.md b/spaces/S000064/properties/P000033.md
diff --git a/spaces/S000064/properties/P000048.md b/spaces/S000064/properties/P000048.md
@@ -3,10 +3,8 @@ space: S000064
 property: P000048
 value: false
 refs:
-- doi: 10.1007/978-1-4612-6290-9_6
+- zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
-If $i$ is irrational, no two points $x_1 = (i, y_1), x_2 = (i, y_2)$ on the vertical line $x = i$ can be separated.
-
-See item #5 for space #72 in {{doi:10.1007/978-1-4612-6290-9_6}}.
+See item #5 for space #72 in {{zb:0386.54001}}.
diff --git a/spaces/S000064/properties/P000051.md b/spaces/S000064/properties/P000051.md
@@ -3,10 +3,10 @@ space: S000064
 property: P000051
 value: true
 refs:
-- doi: 10.1007/978-1-4612-6290-9_6
+- zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
-Since $D$ is discrete, any dense-in-itself subset must be contained in $X - D$; but every point $p \in X - D$ has a neighborhood contained in $\{p\} \cap D$, so no nonempty subset can be dense-in-itself.
+Since $D$ is discrete, any dense-in-itself subset must be contained in $X\setminus D$; but every point $p \in X \setminus D$ has a neighborhood contained in $\{p\} \cup D$, so no nonempty subset can be dense-in-itself.
 
-See item #4 for space #72 in {{doi:10.1007/978-1-4612-6290-9_6}}.
+See item #4 for space #72 in {{zb:0386.54001}}.
diff --git a/spaces/S000064/properties/P000065.md b/spaces/S000064/properties/P000065.md
diff --git a/spaces/S000064/properties/P000082.md b/spaces/S000064/properties/P000082.md
@@ -0,0 +1,10 @@
+---
+space: S000064
+property: P000082
+value: true
+---
+
+For each point $q\in D$, its neighborhood $\{q\}$ is {P53}.
+And for each point $x\in X\setminus D$, its neighborhoood $\{x\}\cup D$ is
+{P203}, {P2} and {P28}, hence {P53}
+[(Explore)](https://topology.pi-base.org/spaces?q=Almost+discrete%2BT1%2BFirst+countable%2B%7EMetrizable).
diff --git a/spaces/S000064/properties/P000093.md b/spaces/S000064/properties/P000093.md
@@ -0,0 +1,7 @@
+---
+space: S000064
+property: P000093
+value: true
+---
+
+By definition every base neighbourhood is countable.
diff --git a/spaces/S000064/properties/P000110.md b/spaces/S000064/properties/P000110.md
@@ -0,0 +1,11 @@
+---
+space: S000064
+property: P000110
+value: true
+---
+
+Fix an enumeration $D=\{q_n:n<\omega\}$ and denote $D_n=\{q_k: k\leq n\}$.
+
+A development for $X$ can be formed out of the open covers
+$\mathscr U_n=\{ U_n(x)\setminus D_n : x\in X\setminus D\}\cup\{\{q\}:q\in D \}$,
+where $U_n(x):=\{x\}\cup\{q\in D:\|q-x\|<\frac1n\}$  and $\|.\|$ is the Euclidean norm.
diff --git a/spaces/S000064/properties/P000118.md b/spaces/S000064/properties/P000118.md
@@ -0,0 +1,14 @@
+---
+space: S000064
+property: P000118
+value: true
+---
+
+A $k$-network $\mathcal N$ can be constructed as the union of the following countably many locally finite families:
+- the family $\{ \{x\}:x\in X\setminus D\}$ of singletons of all points outside $D$, 
+- a family $\{B\}$ for each Euclidean ball $B$ in $D$ belonging to a countable base for the standard topology on  $\mathbb Q^2$,
+- a family $\{\{q\}\}$ for each $q\in D$.
+
+For a compact set $K$ and an open set $U$ with $K\subseteq U$, cover $K$ by a family of open subsets of $U$ consisting of a set $\{x\}\cup B$ (with $B$ as in the second bullet above) for each $x\in K\setminus D$, and the singleton $\{q\}$ for each $q\in K\cap D$. 
+The compact set $K$ is covered by a finite subfamily, so $K \subseteq \bigcup\mathcal{N}^* \subseteq U$
+for some finite $\mathcal{N}^* \subseteq \mathcal{N}$ consisting of finitely many $\{x\}$ with $x\in X\setminus D$, finitely many $B$, and finitely many $\{q\}$ with $q\in D.$
diff --git a/spaces/S000064/properties/P000166.md b/spaces/S000064/properties/P000166.md
@@ -3,8 +3,10 @@ space: S000064
 property: P000166
 value: true
 refs:
-- doi: 10.1007/978-1-4612-6290-9_6
+- zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
-See item #1 for space #72 in {{doi:10.1007/978-1-4612-6290-9_6}}.
+The topology is finer than that of {S176}.
+
+See item #1 for space #72 in {{zb:0386.54001}}.
