From 26c4bbbb51dd70ce90f1e8b509f0a55535974af6 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Mon, 29 Jun 2026 19:09:55 -0400 Subject: [PATCH] LCH typo fix? --- src/topology/main/lch.tex | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/src/topology/main/lch.tex b/src/topology/main/lch.tex index 762c31d..d0a6936 100644 --- a/src/topology/main/lch.tex +++ b/src/topology/main/lch.tex @@ -165,15 +165,14 @@ \begin{proof} $(\bracs{F_E}_{E \in \ce})$: For each $E \in \ce$, $\bracsn{F \in \ce|F \cap \ol E \ne \emptyset}$ is finite by \autoref{lemma:locally-finite-compact}. Let \[ - F_E = \bigcup_{\substack{F \in \ce} \\ F \cap \ol E \ne \emptyset}F + F_E = \bigcup_{\substack{F \in \ce \\ F \cap \ol E \ne \emptyset}}F \] then $F_E \in \cn(\ol{E})$ is relatively compact. Let $N \subset X$ and $E \in \ce$. If $N \cap F_E \ne \emptyset$, then there exists $F \in \ce$ such that $N \cap F \ne \emptyset$ and $F \cap \ol{E} \ne \emptyset$. Thus \begin{align*} - \bracs{E \in \ce|N \cap F_E \ne \emptyset} &\subset \bigcup_{\substack{F \in \ce \\ F \cap N \ne \emptyset}}\bracs{E \in \ce|F \cap \ol{E} \ne \emptyset} \\ - &\subset \bigcup_{\substack{F \in \ce \\ F \cap N \ne \emptyset}}\bracs{E \in \ce|\ol{F} \cap \ol{E} \ne \emptyset} + \bracs{E \in \ce|N \cap F_E \ne \emptyset} &\subset \bigcup_{\substack{F \in \ce \\ F \cap N \ne \emptyset}}\bracs{E \in \ce|F \cap \ol{E} \ne \emptyset} \end{align*}