Complete Radial interval topology

spaces/S000135/README.md

@@ -1,8 +1,6 @@
 ---
 uid: S000135
-name: Radial intervals through the origin of the plane
-aliases:
-- Radial interval topology
+name: Radial interval topology on $\mathbb R^2$
 counterexamples_id: 141
 refs:
 - zb: "0386.54001"

spaces/S000135/properties/P000117.md

@@ -0,0 +1,10 @@
+---
+space: S000135
+property: P000117
+value: true
+refs:
+  - mathse: 5136509
+    name: Is $\mathbb R^2$ with the radial interval topology an $\aleph$-space or a $\sigma$-space?
+---
+
+See {{mathse:5136509}}.

spaces/S000135/properties/P000118.md

@@ -0,0 +1,10 @@
+---
+space: S000135
+property: P000118
+value: false
+refs:
+  - mathse: 5136509
+    name: Is $\mathbb R^2$ with the radial interval topology an $\aleph$-space or a $\sigma$-space?
+---
+
+See {{mathse:5136509}}.

spaces/S000135/properties/P000240.md

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+---
+space: S000135
+property: P000240
+value: true
+---
+
+It is a $1$-dimensional {P240} with $X_0 = \left\{ x \in X \mid \left\|x\right\|_2 \in \mathbb N \right\} \subset X_1 = X$,
+where the sets $1$-cells are segments connecting $(n \cos \theta, n \sin \theta)$ and $((n + 1) \cos \theta, (n + 1) \sin \theta)$ for each $n \in \mathbb N$ and $0 \leq \theta < 2 \pi$.