Left-ray topology on $\omega_1$

spaces/S000217/README.md

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+---
+uid: S000217
+name: Left ray topology on $\omega_1$
+refs:
+  - wikipedia: Alexandrov_topology
+    name: Alexandrov topology on Wikipedia
+---
+
+Let $X=\omega_1$, with a base for the topology consisting of the left rays
+$[0,\alpha]=[0,\alpha+1)$ for $\alpha\in\omega_1$.
+
+The open sets are the downward closed subsets of $X$, which are $X$, $\emptyset$,
+the basic open sets above, and the rays $[0,\alpha)$ with $\alpha$ limit ordinal.
+This is the Alexandrov topology for the reverse ordering on $\omega_1$.

spaces/S000217/properties/P000001.md

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+---
+space: S000217
+property: P000001
+value: true
+---
+
+Given $\alpha<\beta$ in $X$, the open set $[0,\alpha]$ contains $\alpha$ and not $\beta$.

spaces/S000217/properties/P000093.md

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+---
+space: S000217
+property: P000093
+value: true
+---
+
+Each basic open set of the form $[0,\alpha]$ for $\alpha\in\omega_1$ is countable.

spaces/S000217/properties/P000114.md

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+---
+space: S000217
+property: P000114
+value: true
+---
+
+By construction.

spaces/S000217/properties/P000196.md

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+---
+space: S000217
+property: P000196
+value: true
+---
+
+By construction.

spaces/S000217/properties/P000226.md

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+---
+space: S000217
+property: P000226
+value: true
+---
+
+Since the ordinal numbers are well-ordered,
+every nonempty collection of open sets has a minimal element.