From 0f1a0ffcc4c8b8cb81d294dfcdd15b1dd6516231 Mon Sep 17 00:00:00 2001
From: Steven Clontz <steven.clontz@gmail.com>
Date: Wed, 15 Mar 2023 15:59:41 -0500
Subject: [PATCH 1/4] Add results to show that pseudometrizable implies 1st
 countable

---
 properties/P000144.md | 10 ++++++++++
 theorems/T000057.md   |  7 +++----
 theorems/T000326.md   | 12 ++++++++++++
 theorems/T000327.md   | 12 ++++++++++++
 4 files changed, 37 insertions(+), 4 deletions(-)
 create mode 100644 properties/P000144.md
 create mode 100644 theorems/T000326.md
 create mode 100644 theorems/T000327.md

diff --git a/properties/P000144.md b/properties/P000144.md
new file mode 100644
index 0000000000..3ba2337e3f
--- /dev/null
+++ b/properties/P000144.md
@@ -0,0 +1,10 @@
+---
+uid: P000144
+slug: locally-pseudometrizable
+name: Locally pseudometrizable
+refs:
+  - doi: 10.1090/S0002-9947-1969-0251719-4
+    name: TODO
+---
+
+Every point has a neighborhood that is {P121}.
diff --git a/theorems/T000057.md b/theorems/T000057.md
index 6ec3424d17..82dc92ae90 100644
--- a/theorems/T000057.md
+++ b/theorems/T000057.md
@@ -1,7 +1,7 @@
 ---
 uid: T000057
 if:
-  P000121: true
+  P000144: true
 then:
   P000028: true
 refs:
@@ -9,6 +9,5 @@ refs:
   name: Counterexamples in Topology
 ---
 
-If $X$ is pseudometric, for any $x \in X$, $\{B(x,\frac{1}{n})\,|\,n \in \omega\}$ is a countable local base at $x$.
-
-Asserted on page 34-35 of {{doi:10.1007/978-1-4612-6290-9}} for metric spaces.  The same holds for pseudometric spaces.
+If $X$ is {P144}, for any $x \in X$, $\{B(x,\frac{1}{n})\,|\,n \in \omega\}$ is a countable local base at $x$,
+using the local pseduometric for a neighborhood of the point.
diff --git a/theorems/T000326.md b/theorems/T000326.md
new file mode 100644
index 0000000000..d29bfd98cc
--- /dev/null
+++ b/theorems/T000326.md
@@ -0,0 +1,12 @@
+---
+uid: T000326
+if:
+  P000121: true
+then:
+  P000141: true
+refs:
+  - mathse: 4646084
+    name: Unraveling the various definitions of k-space or compactly generated space
+---
+
+See {{mathse:4646084}}.
diff --git a/theorems/T000327.md b/theorems/T000327.md
new file mode 100644
index 0000000000..08fc88cd61
--- /dev/null
+++ b/theorems/T000327.md
@@ -0,0 +1,12 @@
+---
+uid: T000326
+if:
+  P000082: true
+then:
+  P000141: true
+refs:
+  - mathse: 4646084
+    name: Unraveling the various definitions of k-space or compactly generated space
+---
+
+See {{mathse:4646084}}.

From c59a23b569a972424b1557a273defe045dd4a42e Mon Sep 17 00:00:00 2001
From: Steven Clontz <steven.clontz@gmail.com>
Date: Wed, 15 Mar 2023 16:01:13 -0500
Subject: [PATCH 2/4] fix T327 id

---
 theorems/T000327.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/theorems/T000327.md b/theorems/T000327.md
index 08fc88cd61..6ba85ed117 100644
--- a/theorems/T000327.md
+++ b/theorems/T000327.md
@@ -1,5 +1,5 @@
 ---
-uid: T000326
+uid: T000327
 if:
   P000082: true
 then:

From 1672e64206f251521ddf49e228b12d975a75af35 Mon Sep 17 00:00:00 2001
From: Steven Clontz <steven.clontz@gmail.com>
Date: Wed, 15 Mar 2023 16:04:56 -0500
Subject: [PATCH 3/4] fix typos in P144

---
 theorems/T000326.md | 2 +-
 theorems/T000327.md | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/theorems/T000326.md b/theorems/T000326.md
index d29bfd98cc..f91203b7e9 100644
--- a/theorems/T000326.md
+++ b/theorems/T000326.md
@@ -3,7 +3,7 @@ uid: T000326
 if:
   P000121: true
 then:
-  P000141: true
+  P000144: true
 refs:
   - mathse: 4646084
     name: Unraveling the various definitions of k-space or compactly generated space
diff --git a/theorems/T000327.md b/theorems/T000327.md
index 6ba85ed117..2e3001e1cc 100644
--- a/theorems/T000327.md
+++ b/theorems/T000327.md
@@ -3,7 +3,7 @@ uid: T000327
 if:
   P000082: true
 then:
-  P000141: true
+  P000144: true
 refs:
   - mathse: 4646084
     name: Unraveling the various definitions of k-space or compactly generated space

From 7ca0fea56178f19a24b7974f439388c8f6e2b062 Mon Sep 17 00:00:00 2001
From: Steven Clontz <steven.clontz@gmail.com>
Date: Wed, 15 Mar 2023 19:43:59 -0500
Subject: [PATCH 4/4] add mathse reference

---
 theorems/T000057.md | 7 +++----
 theorems/T000326.md | 6 +++---
 theorems/T000327.md | 6 +++---
 3 files changed, 9 insertions(+), 10 deletions(-)

diff --git a/theorems/T000057.md b/theorems/T000057.md
index 82dc92ae90..fe9dbb5187 100644
--- a/theorems/T000057.md
+++ b/theorems/T000057.md
@@ -5,9 +5,8 @@ if:
 then:
   P000028: true
 refs:
-- doi: 10.1007/978-1-4612-6290-9
-  name: Counterexamples in Topology
+  - mathse: 4659902
+    name: How can the implication metrizable -> first countable be generalized?
 ---
 
-If $X$ is {P144}, for any $x \in X$, $\{B(x,\frac{1}{n})\,|\,n \in \omega\}$ is a countable local base at $x$,
-using the local pseduometric for a neighborhood of the point.
+Take a pseduometrizable neighborhood. The balls of rational radius within this neighborhood form a local basis.
diff --git a/theorems/T000326.md b/theorems/T000326.md
index f91203b7e9..9c3486dfc8 100644
--- a/theorems/T000326.md
+++ b/theorems/T000326.md
@@ -5,8 +5,8 @@ if:
 then:
   P000144: true
 refs:
-  - mathse: 4646084
-    name: Unraveling the various definitions of k-space or compactly generated space
+  - mathse: 4659902
+    name: How can the implication metrizable -> first countable be generalized?
 ---
 
-See {{mathse:4646084}}.
+The entire space is the desired neighborhood.
diff --git a/theorems/T000327.md b/theorems/T000327.md
index 2e3001e1cc..6cc4e031e8 100644
--- a/theorems/T000327.md
+++ b/theorems/T000327.md
@@ -5,8 +5,8 @@ if:
 then:
   P000144: true
 refs:
-  - mathse: 4646084
-    name: Unraveling the various definitions of k-space or compactly generated space
+  - mathse: 4659902
+    name: How can the implication metrizable -> first countable be generalized?
 ---
 
-See {{mathse:4646084}}.
+Every metric is a pseduometric.