Completing Warsaw circle (S205)

spaces/S000205/README.md

@@ -6,12 +6,15 @@ refs:
     name: Warsaw circle on Wikipedia
   - doi: 10.1016/j.topol.2013.07.001
     name: Milnor–Thurston homology groups of the Warsaw Circle (J. Przewocki)
+  - mathse: 2423154
+    name: First Betti number of a Reeb graph is not greater than that of the space?
 ---
 
 The set $X=\{(x, \sin\frac{1}{x}):x \in (0,1]\} \cup \{(0,y):y \in [-2,1]\} \cup \{(x,-2):x\in[0,1]\} \cup \{(1,y):y \in [-2,\sin 1]\}$ with the subspace topology inherited from $\mathbb{R}^2$.
 
 It consists of {S114} together with an extra arc joining its rightmost point to the point $(0,-1)$.
-See {{wikipedia:Warsaw_circle}} or section 3 of {{doi:10.1016/j.topol.2013.07.001}}
+See {{wikipedia:Warsaw_circle}} or section 3 of {{doi:10.1016/j.topol.2013.07.001}}.
 
-Note: There are a few closely related but non-isomorphic variants of the *Warsaw circle*.
+Note: There are a few closely related but non-isomorphic variants of the *Warsaw circle*
+(see the pictures in {{mathse:2423154}} for two of them).
 The description chosen here seems to be the most common.

spaces/S000205/properties/P000089.md

@@ -0,0 +1,10 @@
+---
+space: S000205
+property: P000089
+value: true
+refs:
+- mathse: 4775855
+  name: Warsaw circle has the fixed point property
+---
+
+See answer to {{mathse:4775855}}.

spaces/S000205/properties/P000204.md

@@ -0,0 +1,8 @@
+---
+space: S000205
+property: P000204
+value: false
+---
+
+Let $p \in X$. If $p \notin \{0\}\times [-1,1]$, then $X\setminus \{p\}$ is homotopy equivalent to {S114} and {S114|P36}.
+Otherwise $Y = X \setminus (\{0\}\times [-1,1])$ is dense in $X \setminus \{p\}$. Hence $X \setminus \{p\}$ is connected as $Y$ is clearly (path-)connected.