From 30701555debcbb589a2dac795c93596baf34c9b1 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <git@jrpie.de>
Date: Tue, 6 Feb 2024 15:11:31 +0100
Subject: [PATCH] more details

---
 inputs/lecture_03.tex | 8 +++++++-
 1 file changed, 7 insertions(+), 1 deletion(-)

diff --git a/inputs/lecture_03.tex b/inputs/lecture_03.tex
index 85c8c5a..2353482 100644
--- a/inputs/lecture_03.tex
+++ b/inputs/lecture_03.tex
@@ -165,7 +165,13 @@
         Since the cantor space embeds into $X$,
         we get the lower bound.
         Since $X$ is second countable and Hausdorff,
-        we get the upper bound.%
+        we get the upper bound:
+        Let $\langle U_n : n < \omega \rangle$ be a countable basis.
+        Consider the injective function
+        \begin{IEEEeqnarray*}{rCl}
+            f\colon X &\longrightarrow & 2^{ \omega} \\
+             x &\longmapsto & \{n : x \in U_n\}.
+        \end{IEEEeqnarray*}
     }{Lower bound:  $2^{\N} \hookrightarrow X$,
         upper bound: \nth{2} countable and Hausdorff.}
 \end{proof}