From 8c090f3c932482f1997fbfb589a4e7461b78dfb3 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Tue, 22 Sep 2026 10:38:12 -0400
Subject: [PATCH] Slight adjustments.

---
 src/op/example/bounded.tex       | 16 ++++++++++++++++
 src/topology/main/compactify.tex |  4 ++--
 2 files changed, 18 insertions(+), 2 deletions(-)

diff --git a/src/op/example/bounded.tex b/src/op/example/bounded.tex
index f01ad7e..7733455 100644
--- a/src/op/example/bounded.tex
+++ b/src/op/example/bounded.tex
@@ -6,6 +6,22 @@
     Let $H$ be a complex Hilbert space, then $B(H) = L(H; H)$ is the algebra of all bounded linear operators on $H$. 
 \end{definition}
 
+\begin{lemma}
+\label{lemma:unitary-ots}
+    Let $H$ be a complex Hilbert space, then the strong and weak operator topologies on $U(B(H))$ coincide. 
+\end{lemma}
+\begin{proof}
+    Let $S, T \in U(B(H))$ and $\xi \in H$, then
+    \begin{align*}
+        \norm{(S - T)\xi}_H^2 &= \dpn{(S - T)\xi, (S - T)\xi}{H} \\
+        &= \norm{S\xi}_H^2 + \norm{T\xi}_H^2 - 2\text{Re}\angles{S\xi, T\xi}_H \\
+        &= 2\norm{\xi}_H^2  - 2\text{Re}\angles{S\xi, T\xi}_H
+    \end{align*}
+
+    which tends to $0$ as $S \to T$ in the weak operator topology. 
+    
+\end{proof}
+
 
 % 1. Every non-trivial ideal of B(H) contains the finite-rank operators.
 % 2. If H is separable, then the only non-trivial closed idela of B(H) are the compact operators. 
diff --git a/src/topology/main/compactify.tex b/src/topology/main/compactify.tex
index f3dffd3..b331eeb 100644
--- a/src/topology/main/compactify.tex
+++ b/src/topology/main/compactify.tex
@@ -143,7 +143,7 @@
     The pair $(X^*, \iota)$ is the \textbf{one-point compactification} of $X$.
 \end{definition}
 \begin{proof}
-    Let $\infty$ be a point not in $X$, $X^* = X \sqcup \bracs{\infty}$, and $\mathcal{T}^* \subset 2^{X^*}$ such that for each $U \in \mathcal{T}^*$, one of the following holds:
+    (1): Let $\infty$ be a point not in $X$, $X^* = X \sqcup \bracs{\infty}$, and $\mathcal{T}^* \subset 2^{X^*}$ such that for each $U \in \mathcal{T}^*$, one of the following holds:
     \begin{enumerate}[label=(\alph*)]
         \item $U \in \mathcal{T}$.
         \item $\infty \in U$ and $U^c \subset X$ is compact. 
@@ -153,7 +153,7 @@
     
     Let $\iota: X \to X^*$ be the inclusion map. For each $U \in \mathcal{T}^*$ satisfying (b), $\iota^{-1}(U) = U \cap X$. Since $U^c \subset X$ is compact, $U \cap X$ is open by \autoref{proposition:compact-closed}, so $\iota \in C(X; X^*)$. By (a), $\iota$ is an embedding. 
 
-    Finally, let
+    (U): Let
     \[
     \varphi^*: Y \to X^* \quad x \mapsto \begin{cases}
         \varphi^{-1}(x) &x \in \varphi(X) \\