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LCH typo fix?

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This commit is contained in:
Bokuan Li
2026-06-29 19:09:55 -04:00
parent ff22fad2f8
commit 26c4bbbb51
1 changed files with 2 additions and 3 deletions
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5
src/topology/main/lch.tex
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@@ -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*}