Completing Circle with two Origins S209 (#1719)

Author: felixpernegger

spaces/S000083/properties/P000216.md

@@ -4,4 +4,4 @@ property: P000216
 value: true
 ---
 
-Similar to the proof that {S83|P145}, using the fact that {S25|P216}.
+$X$ embeds as a subspace of {S209} and {S209|P216}.

spaces/S000209/properties/P000068.md

@@ -0,0 +1,7 @@
+---
+space: S000209
+property: P000068
+value: false
+---
+
+Similar to {S83|P68}.

spaces/S000209/properties/P000216.md

@@ -0,0 +1,12 @@
+---
+space: S000209
+property: P000216
+value: true
+---
+
+It is enough to show that each open set in $X$ is {P30}.
+The whole space is {P30} since {S209|P16}.
+And a proper open subset of $X$ is a disjoint union of open sets,
+each homeomorphic to either {S170}, {S25} or {S83}.
+Observe each of {S170|P30}, {S25|P30} and {S83|P30}.
+Since the {P30} property is preserved by taking disjoint unions, the result follows.