Some missing traits for Radial intervals at 0 (S135) (#1191)

spaces/S000135/README.md

@@ -8,7 +8,7 @@ refs:
 - doi: 10.1007/978-1-4612-6290-9
   name: Counterexamples in Topology
 ---
-The radial interval topology on the plane $\mathbb{R}^2$ is generated by the basis consisting of open intervals disjoint from the origin on lines passing through the origin, along with sets of the form $\cup\{I_\theta : 0 \leq \theta < \pi\}$ where $I_\theta$ is an open interval about the origin on the line with slope $\tan\theta$.
+The radial interval topology on the plane $X=\mathbb{R}^2$ is generated by the basis consisting of open intervals disjoint from the origin on lines passing through the origin, along with sets of the form $\bigcup\{I_\theta : 0 \leq \theta < \pi\}$ where $I_\theta$ is an open interval about the origin on the line with slope $\tan\theta$.
 
 Defined as counterexample #141 ("Radial Interval Topology")
 in {{doi:10.1007/978-1-4612-6290-9}}.

spaces/S000135/properties/P000008.md → spaces/S000135/properties/P000030.md

@@ -1,10 +1,10 @@
 ---
 space: S000135
-property: P000008
+property: P000030
 value: true
 refs:
 - doi: 10.1007/978-1-4612-6290-9_6
   name: Counterexamples in Topology
 ---
 
-See item #2 for space #141 in {{doi:10.1007/978-1-4612-6290-9_6}}.
+See item #5 for space #141 in {{doi:10.1007/978-1-4612-6290-9_6}}

spaces/S000135/properties/P000043.md

@@ -7,5 +7,4 @@ refs:
   name: Counterexamples in Topology
 ---
 
-Asserted in the General Reference Chart for space #141 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+A basic neighbourhood is either a Euclidean segment or the union of a family of such segments with nonempty intersection.

spaces/S000135/properties/P000080.md

@@ -0,0 +1,13 @@
+---
+space: S000135
+property: P000080
+value: true
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+Take $A\subseteq X$ and $p\in \overline A\setminus A$. If $p\neq(0,0)$ then it admits a countable base of neighbourhoods and picking elements of $A$ from each of those neighbourhoods we obtain the required sequence.
+For $p=(0,0)$, there has to exist at least one ray
+containing elements of $A$ arbitrarily close to $p$. Otherwise a neighbourhood of the origin disjoint from $A$ could be found.
+Hence there exists a sequence in $A$ converging to $p$.

spaces/S000135/properties/P000086.md

@@ -0,0 +1,11 @@
+---
+space: S000135
+property: P000086
+value: false
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+{S135|P23}. But every point
+except the origin has a locally compact neighbourhood (homeomorphic to {S25}).

spaces/S000135/properties/P000089.md

@@ -0,0 +1,14 @@
+---
+space: S000135
+property: P000089
+value: false
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+Consider the map $f:X\to X$ given by
+- $f((x,0))= (x+1,0)$ if $x\geq 0$
+- $f((x,y))= (1,0)$ if $y\neq 0$ or $x\leq 0$.
+
+$f$ is continuous but has no fixed point.

spaces/S000135/properties/P000132.md

@@ -0,0 +1,18 @@
+---
+space: S000135
+property: P000132
+value: true
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+Since $X\setminus\{(0,0)\}$ is homeomorphic to the disjoint union of copies of the real line
+and {S25|P132}, every open set in $X$ missing the origin is an $F_\sigma$.
+It remains to show that the origin has a basis of open $F_\sigma$ neighborhoods.
+Consider $U=\bigcup\{(-\varepsilon_\theta,\varepsilon_\theta)p_\theta: 0\leq \theta< \pi\}$, with $\varepsilon_\theta>0$
+and $p_\theta=(\cos\theta,\sin\theta)$ for each angle $0\leq \theta < \pi$. Then the set
+$V:=\bigcup\{[-\varepsilon_\theta,\varepsilon_\theta]p_\theta: 0\leq \theta< \pi\}$ is closed
+and $U=\bigcup_{n=2}^\infty (1-1/n)V$.
+
+Therefore every open subset of $X$ can be represented as a union of two $F_\sigma$ sets, so it is $F_\sigma$ as well.

spaces/S000135/properties/P000166.md

@@ -0,0 +1,7 @@
+---
+space: S000135
+property: P000166
+value: true
+---
+
+The Euclidean topology on the plane is coarser than the topology on $X$.

spaces/S000135/properties/P000187.md

@@ -0,0 +1,8 @@
+---
+space: S000135
+property: P000187
+value: false
+---
+
+P2 has a winning strategy at $(0,0)$. It is enough to pick
+each of the points $p_n$ on a different line passing through the origin. Then there exists a neighbourgood of $(0,0)$ not containing any of those points.

spaces/S000135/properties/P000198.md

@@ -0,0 +1,12 @@
+---
+space: S000135
+property: P000198
+value: false
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+This topology is finer than {S134},
+(cf. item #1 for space #141 in {{doi:10.1007/978-1-4612-6290-9_6}})
+and {S134|P198}.

spaces/S000135/properties/P000199.md

@@ -0,0 +1,7 @@
+---
+space: S000135
+property: P000199
+value: true
+---
+
+The map $X\times[0,1] \ni (\vec x,t)\mapsto (1-t)\vec x\in X$ is a homotopy between the identity and a constant map.

spaces/S000135/properties/P000205.md

@@ -0,0 +1,16 @@
+---
+space: S000135
+property: P000205
+value: true
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+$X$ is {P36}, for example because {S135|P199}.
+
+For every $p\in X\setminus\{(0,0)\}$ the proper subset
+$\{\lambda p: \lambda >1\}$ is clopen in $X\setminus\{p\}$.
+In the subspace $X\setminus\{(0,0)\}$ every half-line
+starting at the origin is clopen.
+Hence for every $p\in X$ the space $X\setminus\{p\}$ is not connected.

spaces/S000135/properties/P000206.md

@@ -0,0 +1,12 @@
+---
+space: S000135
+property: P000206
+value: true
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+Observe that there are two cases: either the origin belongs to $U_n$ for every $n< \omega$ and the game is lost for the first player or for some $n$, the origin is not in $U_n$. Then the second player can pick $V_n$ contained in a line passing through the origin.
+After that, the game is played on the real line and
+{S25|P206}.