From 65b782e2260c8b785abb390bce2aefc4d6a2a47e Mon Sep 17 00:00:00 2001
From: Batixx <s59fpern@uni-bonn.de>
Date: Fri, 9 Jan 2026 19:53:30 +0100
Subject: [PATCH 1/5] definition + basic properties

---
 spaces/S000217/README.md             | 10 ++++++++++
 spaces/S000217/properties/P000001.md |  7 +++++++
 spaces/S000217/properties/P000093.md |  7 +++++++
 spaces/S000217/properties/P000114.md |  7 +++++++
 spaces/S000217/properties/P000196.md |  7 +++++++
 spaces/S000217/properties/P000216.md |  7 +++++++
 6 files changed, 45 insertions(+)
 create mode 100644 spaces/S000217/README.md
 create mode 100644 spaces/S000217/properties/P000001.md
 create mode 100644 spaces/S000217/properties/P000093.md
 create mode 100644 spaces/S000217/properties/P000114.md
 create mode 100644 spaces/S000217/properties/P000196.md
 create mode 100644 spaces/S000217/properties/P000216.md

diff --git a/spaces/S000217/README.md b/spaces/S000217/README.md
new file mode 100644
index 000000000..059b846ec
--- /dev/null
+++ b/spaces/S000217/README.md
@@ -0,0 +1,10 @@
+---
+uid: S000166
+name: Left ray topology on $\omega_1$
+refs:
+  - wikipedia: Alexandrov_topology
+    name: Alexandrov topology on Wikipedia
+---
+
+$\omega_1$ with the topology of left rays
+$\{\omega_1\}\cup\{(\leftarrow,n):n<\omega_1\} = \{\emptyset,\omega_1\}\cup\{(\leftarrow,n]:n<\omega_1\}$.
diff --git a/spaces/S000217/properties/P000001.md b/spaces/S000217/properties/P000001.md
new file mode 100644
index 000000000..fc7eb8d05
--- /dev/null
+++ b/spaces/S000217/properties/P000001.md
@@ -0,0 +1,7 @@
+---
+space: S000217
+property: P000001
+value: true
+---
+
+Given $m<n$, $(\leftarrow,m]$ is a neighborhood of $m$ missing $n$.
diff --git a/spaces/S000217/properties/P000093.md b/spaces/S000217/properties/P000093.md
new file mode 100644
index 000000000..fc3f0a6fd
--- /dev/null
+++ b/spaces/S000217/properties/P000093.md
@@ -0,0 +1,7 @@
+---
+space: S000217
+property: P000093
+value: true
+---
+
+$\{(\leftarrow,n):n<\omega_1\}$ is a basis of $X$ where each member is countable.
diff --git a/spaces/S000217/properties/P000114.md b/spaces/S000217/properties/P000114.md
new file mode 100644
index 000000000..1ebdc2ca2
--- /dev/null
+++ b/spaces/S000217/properties/P000114.md
@@ -0,0 +1,7 @@
+---
+space: S000217
+property: P000114
+value: true
+---
+
+By construction.
diff --git a/spaces/S000217/properties/P000196.md b/spaces/S000217/properties/P000196.md
new file mode 100644
index 000000000..94c332189
--- /dev/null
+++ b/spaces/S000217/properties/P000196.md
@@ -0,0 +1,7 @@
+---
+space: S000217
+property: P000196
+value: true
+---
+
+By construction.
diff --git a/spaces/S000217/properties/P000216.md b/spaces/S000217/properties/P000216.md
new file mode 100644
index 000000000..446652600
--- /dev/null
+++ b/spaces/S000217/properties/P000216.md
@@ -0,0 +1,7 @@
+---
+space: S000217
+property: P000226
+value: true
+---
+
+Since the ordinal numbers are well-ordered, every collection of open sets has a minimal element.

From 637a3bf767f5e8efc7de442c9c9733001a6072c0 Mon Sep 17 00:00:00 2001
From: Batixx <s59fpern@uni-bonn.de>
Date: Fri, 9 Jan 2026 19:56:09 +0100
Subject: [PATCH 2/5] typo in file name

---
 spaces/S000217/properties/{P000216.md => P000226.md} | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
 rename spaces/S000217/properties/{P000216.md => P000226.md} (100%)

diff --git a/spaces/S000217/properties/P000216.md b/spaces/S000217/properties/P000226.md
similarity index 100%
rename from spaces/S000217/properties/P000216.md
rename to spaces/S000217/properties/P000226.md

From 779a46570fe2c05a734a7e2554bacf5fc89496b0 Mon Sep 17 00:00:00 2001
From: Felix Pernegger <s59fpern@uni-bonn.de>
Date: Fri, 9 Jan 2026 19:57:33 +0100
Subject: [PATCH 3/5] typo

---
 spaces/S000217/README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/spaces/S000217/README.md b/spaces/S000217/README.md
index 059b846ec..499852f39 100644
--- a/spaces/S000217/README.md
+++ b/spaces/S000217/README.md
@@ -1,5 +1,5 @@
 ---
-uid: S000166
+uid: S000217
 name: Left ray topology on $\omega_1$
 refs:
   - wikipedia: Alexandrov_topology

From 7fbcadf45b86e7e6c2496b1290d9c22f17fa5072 Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Thu, 15 Jan 2026 21:26:15 -0500
Subject: [PATCH 4/5] updates

---
 spaces/S000217/README.md             | 8 ++++++--
 spaces/S000217/properties/P000001.md | 2 +-
 spaces/S000217/properties/P000093.md | 2 +-
 spaces/S000217/properties/P000226.md | 3 ++-
 4 files changed, 10 insertions(+), 5 deletions(-)

diff --git a/spaces/S000217/README.md b/spaces/S000217/README.md
index 499852f39..aaedc7335 100644
--- a/spaces/S000217/README.md
+++ b/spaces/S000217/README.md
@@ -6,5 +6,9 @@ refs:
     name: Alexandrov topology on Wikipedia
 ---
 
-$\omega_1$ with the topology of left rays
-$\{\omega_1\}\cup\{(\leftarrow,n):n<\omega_1\} = \{\emptyset,\omega_1\}\cup\{(\leftarrow,n]:n<\omega_1\}$.
+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 $\omega_1$.
diff --git a/spaces/S000217/properties/P000001.md b/spaces/S000217/properties/P000001.md
index fc7eb8d05..3efba79c7 100644
--- a/spaces/S000217/properties/P000001.md
+++ b/spaces/S000217/properties/P000001.md
@@ -4,4 +4,4 @@ property: P000001
 value: true
 ---
 
-Given $m<n$, $(\leftarrow,m]$ is a neighborhood of $m$ missing $n$.
+Given $\alpha<\beta$ in $X$, the open set $[0,\alpha]$ contains $\alpha$ and not $\beta$.
diff --git a/spaces/S000217/properties/P000093.md b/spaces/S000217/properties/P000093.md
index fc3f0a6fd..a545e21d5 100644
--- a/spaces/S000217/properties/P000093.md
+++ b/spaces/S000217/properties/P000093.md
@@ -4,4 +4,4 @@ property: P000093
 value: true
 ---
 
-$\{(\leftarrow,n):n<\omega_1\}$ is a basis of $X$ where each member is countable.
+Each basic open set is countable.
diff --git a/spaces/S000217/properties/P000226.md b/spaces/S000217/properties/P000226.md
index 446652600..5c6b467a9 100644
--- a/spaces/S000217/properties/P000226.md
+++ b/spaces/S000217/properties/P000226.md
@@ -4,4 +4,5 @@ property: P000226
 value: true
 ---
 
-Since the ordinal numbers are well-ordered, every collection of open sets has a minimal element.
+Since the ordinal numbers are well-ordered,
+every nonempty collection of open sets has a minimal element.

From a9f20cb3ee51704f15e9220d7771f6985b4a8e0d Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Fri, 16 Jan 2026 18:24:34 -0500
Subject: [PATCH 5/5] P93 modify

---
 spaces/S000217/properties/P000093.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/spaces/S000217/properties/P000093.md b/spaces/S000217/properties/P000093.md
index a545e21d5..e23fd78b6 100644
--- a/spaces/S000217/properties/P000093.md
+++ b/spaces/S000217/properties/P000093.md
@@ -4,4 +4,4 @@ property: P000093
 value: true
 ---
 
-Each basic open set is countable.
+Each basic open set of the form $[0,\alpha]$ for $\alpha\in\omega_1$ is countable.