From 00847bbbffab734333815e4d79ce04a439f149db Mon Sep 17 00:00:00 2001
From: danflapjax <12000932+danflapjax@users.noreply.github.com>
Date: Thu, 21 Nov 2024 03:50:40 -0800
Subject: [PATCH 1/4] S44 has P201

---
 spaces/S000044/properties/P000086.md | 8 --------
 spaces/S000044/properties/P000201.md | 7 +++++++
 2 files changed, 7 insertions(+), 8 deletions(-)
 delete mode 100644 spaces/S000044/properties/P000086.md
 create mode 100644 spaces/S000044/properties/P000201.md

diff --git a/spaces/S000044/properties/P000086.md b/spaces/S000044/properties/P000086.md
deleted file mode 100644
index 2c23509426..0000000000
--- a/spaces/S000044/properties/P000086.md
+++ /dev/null
@@ -1,8 +0,0 @@
----
-space: S000044
-property: P000086
-value: false
----
-
-$\frac{2}{5}$ belongs to a single closed set $(0,1)$, but
-$\frac{3}{5}$ belongs to two closed sets $(0,1)$ and $[1/2,1)$.
diff --git a/spaces/S000044/properties/P000201.md b/spaces/S000044/properties/P000201.md
new file mode 100644
index 0000000000..bed577615a
--- /dev/null
+++ b/spaces/S000044/properties/P000201.md
@@ -0,0 +1,7 @@
+---
+space: S000044
+property: P000201
+value: true
+---
+
+$\frac{1}{4}$ lies in every open set.

From 41f8d335d1fa5e98b779767bf84b47981826e218 Mon Sep 17 00:00:00 2001
From: Steven Clontz <steven.clontz@gmail.com>
Date: Tue, 26 Nov 2024 19:35:12 -0600
Subject: [PATCH 2/4] add @GeoffreySangston's improvement

---
 spaces/S000044/properties/P000192.md | 10 ++++++++++
 spaces/S000044/properties/P000201.md |  7 -------
 2 files changed, 10 insertions(+), 7 deletions(-)
 create mode 100644 spaces/S000044/properties/P000192.md
 delete mode 100644 spaces/S000044/properties/P000201.md

diff --git a/spaces/S000044/properties/P000192.md b/spaces/S000044/properties/P000192.md
new file mode 100644
index 0000000000..d3f369bcfc
--- /dev/null
+++ b/spaces/S000044/properties/P000192.md
@@ -0,0 +1,10 @@
+---
+space: S000044
+property: P000192
+value: true
+---
+
+The non-empty closed subsets of {S44} are the closed intervals 
+$[\frac{n-1}{n}, 1)$, and each is {P201} witnessed by $\frac{n-1}{n}$: the
+proper open subsets of $[\frac{n-1}{n}, 1)$ are $[\frac{n-1}{n}, \frac{N-1}{N})$ for $N > n$,
+and each contains $\frac{n-1}{n}$.
diff --git a/spaces/S000044/properties/P000201.md b/spaces/S000044/properties/P000201.md
deleted file mode 100644
index bed577615a..0000000000
--- a/spaces/S000044/properties/P000201.md
+++ /dev/null
@@ -1,7 +0,0 @@
----
-space: S000044
-property: P000201
-value: true
----
-
-$\frac{1}{4}$ lies in every open set.

From 62e76cadcc0b497442f86e2cb3ee4873b3c0e135 Mon Sep 17 00:00:00 2001
From: Steven Clontz <steven.clontz@gmail.com>
Date: Tue, 26 Nov 2024 19:39:12 -0600
Subject: [PATCH 3/4] define quasi-sober using internal references

---
 properties/P000192.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/properties/P000192.md b/properties/P000192.md
index 19921e9af9..b8586af8b7 100644
--- a/properties/P000192.md
+++ b/properties/P000192.md
@@ -6,8 +6,8 @@ refs:
   name: Sober spaces and sober sets (Echi & Lazaar)
 ---
 
-Every nonempty irreducible closed subset of $X$ is the closure of a point of $X$, not necessarily unique; that is, every nonempty irreducible closed subset has at least one *generic point*.
-Here, a subset of $X$ is called *irreducible* if it is {P39} with the subspace topology.
+Every non-{P137} {P39} closed subspace of $X$ is
+{P201}.
 
 Equivalently, the Kolmogorov quotient of the space is {P73}. (See {T512}.)
 

From 4d4254d293c62d32895a094477ee92db4242babc Mon Sep 17 00:00:00 2001
From: Steven Clontz <steven.clontz@gmail.com>
Date: Tue, 26 Nov 2024 20:22:02 -0600
Subject: [PATCH 4/4] Update spaces/S000044/properties/P000192.md

Co-authored-by: david20000813 <weg002@ucsd.edu>
---
 spaces/S000044/properties/P000192.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/spaces/S000044/properties/P000192.md b/spaces/S000044/properties/P000192.md
index d3f369bcfc..f5806e2194 100644
--- a/spaces/S000044/properties/P000192.md
+++ b/spaces/S000044/properties/P000192.md
@@ -6,5 +6,5 @@ value: true
 
 The non-empty closed subsets of {S44} are the closed intervals 
 $[\frac{n-1}{n}, 1)$, and each is {P201} witnessed by $\frac{n-1}{n}$: the
-proper open subsets of $[\frac{n-1}{n}, 1)$ are $[\frac{n-1}{n}, \frac{N-1}{N})$ for $N > n$,
+nonempty open proper subsets of $[\frac{n-1}{n}, 1)$ are $[\frac{n-1}{n}, \frac{N-1}{N})$ for $N > n$,
 and each contains $\frac{n-1}{n}$.