An update to double origin plane (S66)

spaces/S000066/README.md

@@ -3,14 +3,17 @@ uid: S000066
 name: Double origin plane
 counterexamples_id: 74
 refs:
-- doi: 10.1007/978-1-4612-6290-9 
+- zb: "0386.54001"
   name: Counterexamples in Topology
 - wikipedia: Double_origin_topology
   name: Double origin topology on Wikipedia
 ---
-Let $X$ consist of the set of points of the plane $\mathbb R^{2}$ together with an additional point $0^{\ast}$. Neighborhoods of points other than the origin $0$ and the point $0^{\ast}$ are the usual open sets of $R^{2} - 0$; as a basis of neighborhoods of $0$ and $0^{\ast}$, we take $V_{n}(0) = \{(x,y):x^{2} + y^{2} < \frac{1}{n^2}, y > 0\} \cup \{0\}$ and $V_{n}(0^{\ast}) = \{(x,y):x^{2} + y^{2} < \frac{1}{n^2}, y < 0\} \cup \{0^{\ast}\}$.
+Let $X$ consist of the set of points of the plane $\mathbb R^{2}$ together with an additional point $0^{\ast}$.
+Neighborhoods of points other than the origin $0$ and the point $0^{\ast}$ are the usual open sets of $R^{2}\setminus \{0\}$;
+as a basis of neighborhoods of $0$ and $0^{\ast}$, we take $V_{n}(0) = \{(x,y):x^{2} + y^{2} < \frac{1}{n^2},\, y > 0\} \cup \{0\}$
+and $V_{n}(0^{\ast}) = \{(x,y):x^{2} + y^{2} < \frac{1}{n^2},\, y < 0\} \cup \{0^{\ast}\}$.
 
 Defined as counterexample #74 ("Double Origin Topology")
-in {{doi:10.1007/978-1-4612-6290-9}}. Note that this is *not* a copy
+in {{zb:0386.54001}}. Note that this is *not* a copy
 of {S176} with the origin
 doubled in the sense of {S83}.

spaces/S000066/properties/P000003.md

@@ -3,19 +3,10 @@ space: S000066
 property: P000003
 value: true
 refs:
-- doi: 10.1007/978-1-4612-6290-9_6
+- zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
-For any points $x , y$ other than $0$ and $0^*$ there are neighborhoods $U$ and $V$ separating $x$ and $y$, since {S176} is {P3}.  For $x = 0$ and $y = 0^*$,
+For any points $x , y\in\mathbb R^2\setminus\{0\}$ there are neighborhoods $U$ and $V$ separating $x$ and $y$, since {S176} is {P3}.
+For $x = 0$ and $y = 0^*$ take $U=V_1(0)$ and $V=V_1(0^*)$.
 
-$U = \{ (x, y) \in \mathbb R : x^2 + y^2 < 1, y > 0 \} \cup \{ 0 \}$
-
-and 
-
-$V = \{ ( x,  y) \in \mathbb R : x^2 + y^2 < 1, y < 0 \} \cup \{ 0^* \}$
-
-are open sets that separate $x$ and $y$.
-
-Asserted in the General Reference Chart for space #74 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.

spaces/S000066/properties/P000004.md

@@ -3,8 +3,8 @@ space: S000066
 property: P000004
 value: false
 refs:
-- doi: 10.1007/978-1-4612-6290-9_6
+- zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
-See item #1 for space #74 in {{doi:10.1007/978-1-4612-6290-9_6}}.
+See item #1 for space #74 in {{zb:0386.54001}}.

spaces/S000066/properties/P000010.md

@@ -3,9 +3,10 @@ space: S000066
 property: P000010
 value: true
 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 #74 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+Euclidean balls contained in $\mathbb R^2\setminus\{0\}$
+and canonical base neighbourhoods of $0$ and $0^*$ are
+interiors of their closures.

spaces/S000066/properties/P000017.md

@@ -3,9 +3,8 @@ space: S000066
 property: P000017
 value: true
 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 #74 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+The sets $K_n:=\{p\in\mathbb R^2:\frac1n\leq\|p\|\leq n\}$ for $n\geq 1$ and $\{0,0^*\}$ form a compact cover of $X$.

spaces/S000066/properties/P000022.md

@@ -2,10 +2,7 @@
 space: S000066
 property: P000022
 value: false
-refs:
-- doi: 10.1007/978-1-4612-6290-9_6
-  name: Counterexamples in Topology
 ---
 
-Asserted in the General Reference Chart for space #74 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+The unbounded map $\mathbb R^2\ni (x,y)\mapsto x$ and $0^*\mapsto 0$
+can be readily verified to be continuous.

spaces/S000066/properties/P000027.md

@@ -3,8 +3,8 @@ space: S000066
 property: P000027
 value: true
 refs:
-- doi: 10.1007/978-1-4612-6290-9_6
+- zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
-See item #2 for space #74 in {{doi:10.1007/978-1-4612-6290-9_6}}.
+See item #2 for space #74 in {{zb:0386.54001}}.

spaces/S000066/properties/P000038.md

@@ -3,8 +3,8 @@ space: S000066
 property: P000038
 value: true
 refs:
-- doi: 10.1007/978-1-4612-6290-9_6
+- zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
-See item #3 for space #74 in {{doi:10.1007/978-1-4612-6290-9_6}}.
+See item #3 for space #74 in {{zb:0386.54001}}.

spaces/S000066/properties/P000042.md

@@ -3,7 +3,7 @@ space: S000066
 property: P000042
 value: true
 refs:
-- doi: 10.1007/978-1-4612-6290-9_6
+- zb: "0386.54001"
   name: Counterexamples in Topology
 ---
 
@@ -12,4 +12,4 @@ and $V_{n}(0^{\ast})$ of $0^{\ast}$ is {P37}.
 And the same holds for small enough open balls around other points.
 
 Asserted in the General Reference Chart for space #74 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+{{zb:0386.54001}}.

spaces/S000066/properties/P000082.md

@@ -0,0 +1,7 @@
+---
+space: S000066
+property: P000082
+value: true
+---
+
+Canonical base enighbourhoods have the topology induced from {S176}.

spaces/S000066/properties/P000089.md

@@ -0,0 +1,9 @@
+---
+space: S000066
+property: P000089
+value: false
+---
+
+Consider the map $f:X\to X$ given by
+$f((x,y))=(x/2,-y)$ for $(x,y)\in\mathbb R^2\setminus\{0\}$ and $f(0)=0^*$, $f(0^*)=0$.
+It has no fixed point and can be readily verified to be a homeomorphism.

spaces/S000066/properties/P000110.md

@@ -0,0 +1,10 @@
+---
+space: S000066
+property: P000110
+value: true
+---
+
+For $x\in\mathbb R^2\setminus\{0\}$ we define $U_n(x):=B_e(x,2^{-n})\setminus\{0\}$.
+Moreover, $U_n(0):=\{0\}\cup\{(x,y)\in\mathbb R^2: y>0,\ x^2+y^2<4^{-n}\}$
+and $U_n(0^*):=\{0^*\}\cup\{(x,y)\in\mathbb R^2: y<0,\ x^2+y^2<4^{-n}\}$.
+The families $\mathscr V_n:=\{ U_n(x): x\in X\}$ can be readily verified to form a development.

spaces/S000066/properties/P000206.md

@@ -0,0 +1,15 @@
+---
+space: S000066
+property: P000206
+value: true
+---
+
+The space contains precisely two points with neighbourhoods not homeomorphic to {S176}.
+
+Case 1. If player 1 picks $x_n\in\mathbb R^2\setminus\{0\}$, then
+player 2 wins the game, since {S176|P206}.
+
+Case 2. If $x_n$ is one of the zero points, player 2 can
+choose $V_n$ to exclude the other zero. Then either
+$(x_m)_{m\geq n}$ is constant (and player 2 wins) or at some point we have Case 1.
+