From 1ab81b49bc946961b3d208d3623c26ad9fc76f75 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Mon, 29 Jun 2026 21:07:59 -0400 Subject: [PATCH] RETRACTION ON LCH EMBARRASSING TYPO --- src/topology/main/lch.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/topology/main/lch.tex b/src/topology/main/lch.tex index d0a6936..2096fcc 100644 --- a/src/topology/main/lch.tex +++ b/src/topology/main/lch.tex @@ -160,7 +160,7 @@ \begin{lemma} \label{lemma:lch-locally-finite-relatively compact-refine} - Let $X$ be a LCH space and $\ce \subset 2^X$ be a locally finite relatively compact open cover of $X$, then there exists locally finite relatively compact open covers $\bracs{F_E}_{E \in \ce}, \bracs{G_E}_{E \in \ce} \subset 2^X$ such that for each $E \in \ce$, $F_E \subset \ol{F_E} \subset E \subset \ol{E} \subset G_E$. + Let $X$ be a LCH space and $\ce \subset 2^X$ be a locally finite relatively compact open cover of $X$, then there exists locally finite relatively compact open covers $\bracs{F_E}_{E \in \ce}, \bracs{G_E}_{E \in \ce} \subset 2^X$ such that for each $E \in \ce$, $G_E \subset \ol{G_E} \subset E \subset \ol{E} \subset F_E$. \end{lemma} \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 @@ -176,7 +176,7 @@ \end{align*} - By \autoref{lemma:locally-finite-closure}, $\bracsn{\ol E|E \in \ce}$ is also locally finite. Hence for every $F \in \ce$, $\bracsn{E \in \ce|F \cap \ol{E} \ne \emptyset}$ is finite. + By \autoref{lemma:locally-finite-closure}, $\bracsn{\ol E|E \in \ce}$ is also locally finite. Hence for every $F \in \ce$, $\bracsn{E \in \ce|F_F \cap \ol{E} \ne \emptyset}$ is finite. Let $x \in X$, then there exists $N \in \cn(x)$ such that $\bracs{F \in \ce|N \cap F \ne \emptyset}$ is finite. In which case, $\bracs{E \in \ce|N \cap F_E \ne \emptyset}$ is finite as well. Therefore $\bracs{F_E}_{E \in \ce}$ is locally finite.