From e9b4937142758acec805f3b52313aa1df74ad621 Mon Sep 17 00:00:00 2001
From: Batixx <s59fpern@uni-bonn.de>
Date: Sun, 28 Dec 2025 21:08:05 +0100
Subject: [PATCH 1/5] Toronto spaces itroduction + easy properties

---
 properties/P000219.md | 13 +++++++++++++
 theorems/T000805.md   |  9 +++++++++
 theorems/T000806.md   |  9 +++++++++
 theorems/T000811.md   |  9 +++++++++
 theorems/T000812.md   |  9 +++++++++
 theorems/T000813.md   | 11 +++++++++++
 6 files changed, 60 insertions(+)
 create mode 100644 properties/P000219.md
 create mode 100644 theorems/T000805.md
 create mode 100644 theorems/T000806.md
 create mode 100644 theorems/T000811.md
 create mode 100644 theorems/T000812.md
 create mode 100644 theorems/T000813.md

diff --git a/properties/P000219.md b/properties/P000219.md
new file mode 100644
index 000000000..20f82f96b
--- /dev/null
+++ b/properties/P000219.md
@@ -0,0 +1,13 @@
+---
+uid: P000219
+name: Toronto
+refs:
+  - wikipedia: Toronto_space
+    name: Toronto space on Wikipedia
+  - zb: "1286.54032"
+    name: The Toronto Problem (W. R. Brian)
+---
+
+Any subspace $Y \subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.
+
+In {{zb:1286.54032}} it is shown that under GCH, every {P3} Toronto space is {P52}.
diff --git a/theorems/T000805.md b/theorems/T000805.md
new file mode 100644
index 000000000..a38e04884
--- /dev/null
+++ b/theorems/T000805.md
@@ -0,0 +1,9 @@
+---
+uid: T000805
+if:
+  P000078: true
+then:
+  P000219: true
+---
+
+For a finite space $X$, the only subspace with the same cardinality is $X$ itself, which is trivally homeomorphic to $X$.
diff --git a/theorems/T000806.md b/theorems/T000806.md
new file mode 100644
index 000000000..e99cf3abe
--- /dev/null
+++ b/theorems/T000806.md
@@ -0,0 +1,9 @@
+---
+uid: T000806
+if:
+  P000052: true
+then:
+  P000219: true
+---
+
+Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.
diff --git a/theorems/T000811.md b/theorems/T000811.md
new file mode 100644
index 000000000..4362b9f86
--- /dev/null
+++ b/theorems/T000811.md
@@ -0,0 +1,9 @@
+---
+uid: T000811
+if:
+  P000222: true
+then:
+  P000219: true
+---
+
+Let $Y\subset X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.
diff --git a/theorems/T000812.md b/theorems/T000812.md
new file mode 100644
index 000000000..35456be10
--- /dev/null
+++ b/theorems/T000812.md
@@ -0,0 +1,9 @@
+---
+uid: T000812
+if:
+  P000129: true
+then:
+  P000219: true
+---
+
+Let $Y\subset X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.
diff --git a/theorems/T000813.md b/theorems/T000813.md
new file mode 100644
index 000000000..f03fa3d5c
--- /dev/null
+++ b/theorems/T000813.md
@@ -0,0 +1,11 @@
+---
+uid: T000813
+if:
+  and:
+  - P000219: true
+  - P000204: true
+then:
+  P000078: true
+---
+
+Let $p$ be a cut point of $X$. If $X$ were infinite, $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.

From b620f8408dffc634e75d616333a1ec3409bbd69c Mon Sep 17 00:00:00 2001
From: Batixx <s59fpern@uni-bonn.de>
Date: Sun, 28 Dec 2025 21:13:51 +0100
Subject: [PATCH 2/5] resolve merge conflict

---
 theorems/{T000812.md => T000814.md} | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)
 rename theorems/{T000812.md => T000814.md} (91%)

diff --git a/theorems/T000812.md b/theorems/T000814.md
similarity index 91%
rename from theorems/T000812.md
rename to theorems/T000814.md
index 35456be10..51a8db5a6 100644
--- a/theorems/T000812.md
+++ b/theorems/T000814.md
@@ -1,5 +1,5 @@
 ---
-uid: T000812
+uid: T000814
 if:
   P000129: true
 then:

From 84ec0271fe4ea4a0299ee6e542c3d491ea1945b6 Mon Sep 17 00:00:00 2001
From: Batixx <s59fpern@uni-bonn.de>
Date: Sun, 28 Dec 2025 22:43:18 +0100
Subject: [PATCH 3/5] any -> every typo

---
 properties/P000219.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/properties/P000219.md b/properties/P000219.md
index 20f82f96b..a6326f0e6 100644
--- a/properties/P000219.md
+++ b/properties/P000219.md
@@ -8,6 +8,6 @@ refs:
     name: The Toronto Problem (W. R. Brian)
 ---
 
-Any subspace $Y \subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.
+Every subspace $Y \subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.
 
 In {{zb:1286.54032}} it is shown that under GCH, every {P3} Toronto space is {P52}.

From c6eb101e0a070829b314456ed9590e685aff163d Mon Sep 17 00:00:00 2001
From: Batixx <s59fpern@uni-bonn.de>
Date: Mon, 29 Dec 2025 01:10:53 +0100
Subject: [PATCH 4/5] apply suggestions

---
 theorems/T000813.md                 | 11 -----------
 theorems/T000815.md                 | 11 +++++++++++
 theorems/{T000811.md => T000816.md} |  2 +-
 theorems/{T000806.md => T000817.md} |  2 +-
 theorems/{T000805.md => T000818.md} |  2 +-
 5 files changed, 14 insertions(+), 14 deletions(-)
 delete mode 100644 theorems/T000813.md
 create mode 100644 theorems/T000815.md
 rename theorems/{T000811.md => T000816.md} (91%)
 rename theorems/{T000806.md => T000817.md} (91%)
 rename theorems/{T000805.md => T000818.md} (93%)

diff --git a/theorems/T000813.md b/theorems/T000813.md
deleted file mode 100644
index f03fa3d5c..000000000
--- a/theorems/T000813.md
+++ /dev/null
@@ -1,11 +0,0 @@
----
-uid: T000813
-if:
-  and:
-  - P000219: true
-  - P000204: true
-then:
-  P000078: true
----
-
-Let $p$ be a cut point of $X$. If $X$ were infinite, $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.
diff --git a/theorems/T000815.md b/theorems/T000815.md
new file mode 100644
index 000000000..606c6c983
--- /dev/null
+++ b/theorems/T000815.md
@@ -0,0 +1,11 @@
+---
+uid: T000815
+if:
+  and:
+  - P000219: true
+  - P000204: false
+then:
+  P000078: false
+---
+
+Assume $X$ has a cut point $p$. Then $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.
diff --git a/theorems/T000811.md b/theorems/T000816.md
similarity index 91%
rename from theorems/T000811.md
rename to theorems/T000816.md
index 4362b9f86..75f4923ed 100644
--- a/theorems/T000811.md
+++ b/theorems/T000816.md
@@ -1,5 +1,5 @@
 ---
-uid: T000811
+uid: T000816
 if:
   P000222: true
 then:
diff --git a/theorems/T000806.md b/theorems/T000817.md
similarity index 91%
rename from theorems/T000806.md
rename to theorems/T000817.md
index e99cf3abe..15048d117 100644
--- a/theorems/T000806.md
+++ b/theorems/T000817.md
@@ -1,5 +1,5 @@
 ---
-uid: T000806
+uid: T000817
 if:
   P000052: true
 then:
diff --git a/theorems/T000805.md b/theorems/T000818.md
similarity index 93%
rename from theorems/T000805.md
rename to theorems/T000818.md
index a38e04884..9d0246143 100644
--- a/theorems/T000805.md
+++ b/theorems/T000818.md
@@ -1,5 +1,5 @@
 ---
-uid: T000805
+uid: T000818
 if:
   P000078: true
 then:

From e6f5d6eaae227e47781c82495c094220fbf2d5e4 Mon Sep 17 00:00:00 2001
From: Batixx <s59fpern@uni-bonn.de>
Date: Mon, 29 Dec 2025 01:17:18 +0100
Subject: [PATCH 5/5] apply suggestion

---
 theorems/T000815.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/theorems/T000815.md b/theorems/T000815.md
index 606c6c983..4a667ae98 100644
--- a/theorems/T000815.md
+++ b/theorems/T000815.md
@@ -3,9 +3,9 @@ uid: T000815
 if:
   and:
   - P000219: true
-  - P000204: false
+  - P000078: false
 then:
-  P000078: false
+  P000204: false
 ---
 
 Assume $X$ has a cut point $p$. Then $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.