From 28594d15a2be0e43dc24a8162d1884e9f7a23b32 Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Thu, 2 Jul 2026 17:56:19 -0400
Subject: [PATCH 1/2] T913: cid + not discrete => uncountable pi-character

---
 theorems/T000913.md | 16 ++++++++++++++++
 1 file changed, 16 insertions(+)
 create mode 100644 theorems/T000913.md

diff --git a/theorems/T000913.md b/theorems/T000913.md
new file mode 100644
index 000000000..3c609c511
--- /dev/null
+++ b/theorems/T000913.md
@@ -0,0 +1,16 @@
+---
+uid: T000913
+if:
+  and:
+  - P000168: true
+  - P000052: false
+then:
+  P000244: false
+---
+
+Let $x$ be a non-isolated point in $X$
+and let $\mathscr V$ be a countable collection of nonempty open sets.
+To show $X$ has uncountable $\pi$-character, we show that $\mathscr V$ is not a local $\pi$-base for $x$.
+Each $V\in\mathscr V$ contains a point $x_V\ne x$.
+The set $F:=\{x_V:V\in\mathscr V\}$ is countable, hence closed since $X$ is {P168}.
+Then $X\setminus F$ is an open neighborhood of $x$ not containing any $V\in\mathscr V$.

From e0e742f4b079636d36d481a260918a7e9ca00a26 Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Thu, 2 Jul 2026 23:02:39 -0400
Subject: [PATCH 2/2] clearer

---
 theorems/T000913.md | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/theorems/T000913.md b/theorems/T000913.md
index 3c609c511..dc8c39857 100644
--- a/theorems/T000913.md
+++ b/theorems/T000913.md
@@ -12,5 +12,6 @@ Let $x$ be a non-isolated point in $X$
 and let $\mathscr V$ be a countable collection of nonempty open sets.
 To show $X$ has uncountable $\pi$-character, we show that $\mathscr V$ is not a local $\pi$-base for $x$.
 Each $V\in\mathscr V$ contains a point $x_V\ne x$.
-The set $F:=\{x_V:V\in\mathscr V\}$ is countable, hence closed since $X$ is {P168}.
-Then $X\setminus F$ is an open neighborhood of $x$ not containing any $V\in\mathscr V$.
+One characterization of {P168} is that every countable set is closed.
+So the countable set $F:=\{x_V:V\in\mathscr V\}$ is closed in $X$.
+Its complement $X\setminus F$ is an open neighborhood of $x$ not containing any $V\in\mathscr V$.