Introduce S209 Circle with two origins

spaces/S000209/README.md

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+---
+uid: S000209
+name: Circle with two origins
+refs:
+  - wikipedia: Alexandroff_extension
+    name: Alexandroff extension on Wikipedia
+---
+
+Choose a point $0 \in S^1$ to be called the origin, and replace $0$ with two origins $0_1$ and $0_2$. Basic open neighborhoods of a point $x \neq 0$ are Euclidean open neighborhoods of $x$ not containing $0$. Basic open neighborhoods of each origin $0_i$ are of the form $(U\setminus\{0\})\cup\{0_i\}$ with $U$ a Euclidean open neighborhood of $0$.
+
+Let $\{1, 2\}$ have the discrete topology. $X$ is homeomorphic to the quotient space of $S^1 \times \{1, 2\}$ obtained by identifying $\langle \theta, 1 \rangle$ and $\langle \theta, 2 \rangle$ exactly when $\theta {\not\equiv} 0 \mod 2\pi$.
+
+$X$ is homeomorphic to the Alexandroff extension of {S83}.

spaces/S000209/properties/P000016.md

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+---
+space: S000209
+property: P000016
+value: true
+refs:
+  - wikipedia: Alexandroff_extension
+    name: Alexandroff extension on Wikipedia
+---
+
+$X$ is homeomorphic to the Alexandroff extension of {S83}.

spaces/S000209/properties/P000038.md

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+---
+space: S000209
+property: P000038
+value: true
+---
+
+The map $[0, 2\pi] \to X$ defined by $t \mapsto \langle t, 1 \rangle$ if $t < 2\pi$ and $2\pi \mapsto \langle 2\pi, 2\rangle$ is injective and continuous. It is clear by restricting this map to sub-intervals and reparameterizing the results that every pair of points is connected by an injective path.

spaces/S000209/properties/P000101.md

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+---
+space: S000209
+property: P000101
+value: false
+---
+
+Same argument as {S83|P101}.

spaces/S000209/properties/P000123.md

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+---
+space: S000209
+property: P000123
+value: true
+---
+
+Each point is contained in an open set homeomorphic to $S^1$, namely $X\setminus\{0_1\}$ or $X\setminus\{0_2\}$, and {S170|P123}.

spaces/S000209/properties/P000169.md

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+---
+space: S000209
+property: P000169
+value: false
+---
+
+Same argument as {S83|P169}.

spaces/S000209/properties/P000200.md

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+---
+space: S000209
+property: P000200
+value: false
+refs:
+  - zb: "1044.55001"
+    name: Algebraic Topology (Hatcher)
+---
+
+The map sending the origins to $0_1$ and fixing all other points is a retraction onto {S170}. Since {S170|P200}, it follows by Proposition 1.17 of {{zb:1044.55001}} that $X$ is not simply connected.

spaces/S000209/properties/P000204.md

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+---
+space: S000209
+property: P000204
+value: false
+---
+
+For any $p \in X$, $X \backslash \{p\}$ is either homeomorphic to {S83} or {S170}.