proofs

spaces/S000134/README.md

@@ -10,11 +10,11 @@ refs:
   - wikipedia: Hedgehog_space
     name: Hedgehog space
 ---
-If $d$ is the usual metric on $\mathbb{R}^2$, the radial metric $r$ is defined by:
+If $d_e$ is the usual metric on $\mathbb{R}^2$, the radial metric $r$ is defined by:
 
 $r(x,y) = \begin{cases}
-d(x,y), & x \text{ and } y \text{ colinear with } \vec 0\\
-d(x,\vec 0) + d(y,\vec 0), & \text{otherwise.}
+d_e(x,y), & x \text{ and } y \text{ are colinear with } \vec 0\\
+d_e(x,\vec 0) + d_e(y,\vec 0), & \text{otherwise.}
 \end{cases}$
 
 Defined as counterexample #140 ("The Radial Metric")

spaces/S000134/properties/P000042.md

@@ -0,0 +1,18 @@
+---
+space: S000134
+property: P000042
+value: true
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+Every open half line emanating from the origin is
+isometric to {S25}
+and {S25|P42}.
+
+It remains to show that the origin admits a local base of path connected neighbourhoods.
+
+A ball centered at zero $B_r(\vec 0, \varepsilon)$ coincides with the Euclidean ball. Every
+point $\vec x\in B_r(\vec 0, \varepsilon)$ can be joined with the origin
+by a path $t\mapsto (1-t)\vec x$.

spaces/S000134/properties/P000055.md

@@ -7,5 +7,11 @@ refs:
   name: Counterexamples in Topology
 ---
 
-Asserted in the General Reference Chart for space #140 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+The metric $r$ is complete.
+Every line passing through the origin is isometric to {S25},
+hence a Cauchy sequence eventually contained in such a line has a limit by completeness
+of the Euclidean metric.
+
+Assume $(x_n)$ is a Cauchy sequence not contained in a line, i.e. for
+every $n$ there exists $n'>n$ such that $x_n$ and $x_{n'}$ are not colinear with $\vec 0$.
+Then $r(x_n,x_{n'}) = d_e(x_n,\vec 0)+d_e(x_{n'},\vec 0)\geq r(x_n,\vec 0)$. The assumption $r(x_n,x_m)\to 0$ (for $n,m\to \infty$) implies $r(x_n,\vec 0)\to 0$, hence the sequence has a limit.