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Fixed typo in LCH.
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Bokuan Li
2026-06-23 15:11:01 -04:00
parent ea097e46e8
commit 15dec0e93f
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src/topology/main/lch.tex
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@@ -48,7 +48,7 @@
As $\ol{W}$ is compact, it is normal by \autoref{proposition:compact-hausdorff-normal}. Since $X$ is Hausdorff, $K \subset \ol{W}$ is closed by \autoref{proposition:compact-closed}. As $\ol{W}$ is compact, it is normal by \autoref{proposition:compact-hausdorff-normal}. Since $X$ is Hausdorff, $K \subset \ol{W}$ is closed by \autoref{proposition:compact-closed}.
By \hyperref[Urysohn's lemma]{lemma:urysohn}, there exists $f \in C(\ol{V}; [0, 1])$ such that $f|_K = 1$ and $f|_{\ol{W} \setminus V} = 0$. Let By \hyperref[Urysohn's lemma]{lemma:urysohn}, there exists $f \in C(\ol{W}; [0, 1])$ such that $f|_K = 1$ and $f|_{\ol{W} \setminus V} = 0$. Let
\[ \[
F: X \to [0, 1] \quad x \mapsto \begin{cases} F: X \to [0, 1] \quad x \mapsto \begin{cases}
f(x) &x \in W \\ f(x) &x \in W \\