From 3f145bbf9fab613678ad9fb1832be47caea02679 Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Thu, 13 Aug 2026 03:44:37 -0400
Subject: [PATCH 1/2] Spaces with a dispersion point are not cut point spaces

---
 theorems/T000928.md | 13 +++++++------
 1 file changed, 7 insertions(+), 6 deletions(-)

diff --git a/theorems/T000928.md b/theorems/T000928.md
index ae3a8be96..05f73c8f2 100644
--- a/theorems/T000928.md
+++ b/theorems/T000928.md
@@ -1,15 +1,16 @@
 ---
 uid: T000928
 if:
-  and:
-  - P000045: true
-  - P000175: true
+  P000045: true
 then:
-  P000086: false
+  P000205: false
 refs:
   - mathse: 5146777
     name: Answer to "A connected space cannot have more than one dispersion point"
 ---
 
-If $|X|\ge 3$, $X$ can have at most one dispersion point.
-See Proposition 1 in {{mathse:5146777}}.
+Let $p$ be a dispersion point of $X$. If $|X|\ge 2$ and $q$ is a point distinct from $p$,
+the set $X\setminus\{q\}$ is connected by Proposition 2 in {{mathse:5146777}},
+so $q$ is not a cut point.
+And if $|X|\le 1$, $X$ has no cut point since {T558}.
+In both cases $X$ is not a {P205}.

From f1febd6ec501527752d5322588710263e2130918 Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Thu, 13 Aug 2026 16:18:34 -0400
Subject: [PATCH 2/2] trait cleanup

---
 spaces/S000008/properties/P000205.md | 7 -------
 spaces/S000009/properties/P000205.md | 8 --------
 2 files changed, 15 deletions(-)
 delete mode 100644 spaces/S000008/properties/P000205.md
 delete mode 100644 spaces/S000009/properties/P000205.md

diff --git a/spaces/S000008/properties/P000205.md b/spaces/S000008/properties/P000205.md
deleted file mode 100644
index 8899a5c37..000000000
--- a/spaces/S000008/properties/P000205.md
+++ /dev/null
@@ -1,7 +0,0 @@
----
-space: S000008
-property: P000205
-value: false
----
-
-If $x\neq p$ then any bijection $f:X\setminus\{x\}\to X$ with $f(p) = p$ is a homeomorphism, and {S8|P36}, so no $x\in X\setminus \{p\}$ is a cut point.
diff --git a/spaces/S000009/properties/P000205.md b/spaces/S000009/properties/P000205.md
deleted file mode 100644
index d0876ebf9..000000000
--- a/spaces/S000009/properties/P000205.md
+++ /dev/null
@@ -1,8 +0,0 @@
----
-space: S000009
-property: P000205
-value: false
----
-
-Removing any point other than the particular point from $X$ results in a space homeomorphic to $X$
-and {S9|P36}.