Small adjustments.

src/topology/main/lch.tex

@@ -188,14 +188,14 @@
         \item[(b)] For every $x \in X_E$, $N_x \cap E \ne \emptyset$.
     \end{enumerate}
 
-    Let $X_{\ce} = \bigcup_{E \in \ce}X_\ce$, and for each $E \in \ce$, let
+    Let $X_{\ce} = \bigcup_{E \in \ce}X_E$, and for each $E \in \ce$, let
     \[
     G_E = \bigcup_{\substack{x \in X_\ce \\ \ol{N_x} \subset E}}N_x
     \]
 
     then $\bracs{G_E}_{E \in \ce}$ is an open cover of $X$. Since $G_E \subset E$ for all $E \in \ce$, $\bracs{G_E}_{E \in \ce}$ is locally finite.
 
-    It remains to show that $\ol{G_E} \subset E$. Let $x \in X_F$ such that $N_x \subset E$, then $N_x \cap F \ne \emptyset$. Since $N_x \subset E$, $E \cap F \ne \emptyset$. Thus
+    It remains to show that $\ol{G_E} \subset E$. Let $F \in \ce$ and $x \in X_F$ such that $N_x \subset E$, then $N_x \cap F \ne \emptyset$ and $E \cap F \ne \emptyset$. Thus
     \[
     \bracsn{x \in X_\ce|\ol{N_x} \subset E} \subset \bigcup_{\substack{F \in \ce \\ E \cap F \ne \emptyset}}X_F \subset \bigcup_{\substack{F \in \ce \\ \ol E \cap F \ne \emptyset}}X_F
     \]