Adjustments added to LCH spaces.

src/topology/main/lch.tex

@@ -163,7 +163,7 @@
     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 $E \in \ce$, $F_E \subset \ol{F_E} \subset E \subset \ol{E} \subset G_E$.
 \end{lemma}
 \begin{proof}
-    $(\bracs{F_E}_{E \in \ce})$: For each $E \in \ce$, $\bracs{F \in \ce|F \cap \ol E \ne \emptyset}$ is finite by \autoref{lemma:locally-finite-compact}. Let
+    $(\bracs{F_E}_{E \in \ce})$: For each $E \in \ce$, $\bracsn{F \in \ce|F \cap \ol E \ne \emptyset}$ is finite by \autoref{lemma:locally-finite-compact}. Let
     \[
     F_E = \bigcup_{\substack{F \in \ce} \\ F \cap \ol E \ne \emptyset}F
     \]
@@ -171,9 +171,11 @@
     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
-    \[
-    \bracs{E \in \ce|N \cap F_E \ne \emptyset} \subset \bigcup_{\substack{F \in \ce \\ F \cap N \ne \emptyset}}\bracs{E \in \ce|F \cap \ol{E} \ne \emptyset} \subset \bigcup_{\substack{F \in \ce \\ F \cap N \ne \emptyset}}\bracs{E \in \ce|\ol{F} \cap \ol{E} \ne \emptyset}
-    \]
+    \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}
+    \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.
 
@@ -207,8 +209,8 @@
     by \autoref{proposition:closure-finite-union}.
 \end{proof}
 
-\begin{proposition}
-\label{proposition:lch-paracompact}
+\begin{theorem}
+\label{theorem:lch-paracompact}
     Let $X$ be a LCH space, then the following are equivalent:
     \begin{enumerate}
         \item $X$ is paracompact.
@@ -218,7 +220,7 @@
         \item For any open cover $\mathcal{U}$ of $X$, there exists a $C_c(X; [0, 1])$ partition of unity subordinate to it.
         \item $X$ admits a $C_c(X; [0, 1])$ partition of unity.
     \end{enumerate}
-\end{proposition}
+\end{theorem}
 \begin{proof}
     (1) $\Rightarrow$ (2): For each $x \in X$, there exists a relatively compact open neighbourhood $U_x \in \cn^o(x)$. Since $\bracs{U_x| x \in X}$ is an open cover of $X$, there exists a locally finite refinement $\mathcal{V}$. For each $V \in \mathcal{V}$, there exists $x \in X$ such that $V \subset U_x$. In which case, $\ol{V} \subset \ol{U_x}$ is compact.
 
@@ -231,9 +233,9 @@
 
     then $\mathcal{U}_F$ is a relatively compact open cover of $\ol{F}$. By compactness of $\ol{F}$, there exists $\mathcal{V}_F \subset \mathcal{U}_F$ finite such that $\ol{F} \subset \bigcup_{V \in \mathcal{V}_F}V$.
 
-    Let $\mathcal{V} = \bigcup_{F \in \cf}\mathcal{V}_F$, then $\mathcal{V}$ is a relatively compact open cover of $X$. For any $x \in X$, there exists $N \in \cn(x)$ such that $\bracs{F \in \cf|N \cap G_F}$ is finite. Thus
+    Let $\mathcal{V} = \bigcup_{F \in \cf}\mathcal{V}_F$, then $\mathcal{V}$ is a relatively compact open cover of $X$. For any $x \in X$, there exists $N \in \cn(x)$ such that $\bracs{F \in \cf|N \cap G_F \ne \emptyset}$ is finite. Thus
     \[
-    \bracs{V \in \mathcal{V}| N \cap V} \subset \bigcup_{\substack{F \in \cf \\ N \cap G_F \ne \emptyset}}\mathcal{V}_F
+    \bracs{V \in \mathcal{V}| N \cap V \ne \emptyset} \subset \bigcup_{\substack{F \in \cf \\ N \cap G_F \ne \emptyset}}\mathcal{V}_F
     \]
 
     is finite, and $\mathcal{V}$ is locally finite.
@@ -260,21 +262,23 @@
     (6) $\Rightarrow$ (2): Let $\seqi{f} \subset C_c(X; [0, 1])$ be a partition of unity. For each $i \in I$, let $V_i = \bracs{f_i > 0}$, then $\seqi{V}$ is a locally finite relatively compact open cover of $\mathcal{U}$.
 \end{proof}
 
-\begin{proposition}
-\label{proposition:paracompact-lch-normal}
+There is a more general fact that paracompact Hausdorff spaces are normal \cite[Theorem 6.41.1]{Munkres}. However, the above characterisation can be abused to quickly obtain the same result with local compactness. 
+
+\begin{corollary}
+\label{corollary:paracompact-lch-normal}
     Let $X$ be a paracompact LCH space, then $X$ is normal. 
-\end{proposition}
+\end{corollary}
 \begin{proof}
-    Let $A, B \subset X$ be disjoint closed sets. By \autoref{proposition:lch-paracompact}, there exists a partition of unity $\seqi{f}$ subordinate to $\bracs{A^c, B^c}$. Let $I = I_A \sqcup I_B$ such that for each $i \in I_A$, $\supp{f_i} \subset B^c$, and for each $i \in I_B$, $\supp{f_i} \subset A^c$. Take $f = \sum_{i \in I_A}f_i$ and $g = \sum_{i \in I_B}f_i$, then since $f|_B = 0$ and $g|_A = 0$, $\bracs{f \ge 2/3} \in \cn_X(A)$ and $\bracs{f \le 1/3} \in \cn_X(B)$. 
+    Let $A, B \subset X$ be disjoint closed sets. By \autoref{theorem:lch-paracompact}, there exists a partition of unity $\seqi{f}$ subordinate to $\bracs{A^c, B^c}$. Let $I = I_A \sqcup I_B$ such that for each $i \in I_A$, $\supp{f_i} \subset B^c$, and for each $i \in I_B$, $\supp{f_i} \subset A^c$. Take $f = \sum_{i \in I_A}f_i$ and $g = \sum_{i \in I_B}f_i$, then since $f|_B = 0$ and $g|_A = 0$, $\bracs{f \ge 2/3} \in \cn_X(A)$ and $\bracs{f \le 1/3} \in \cn_X(B)$. 
 \end{proof}
 
-\begin{proposition}
-\label{proposition:lch-sigma-paracompact}
+\begin{corollary}
+\label{corollary:lch-sigma-paracompact}
     Let $X$ be a $\sigma$-compact LCH space, then $X$ is paracompact.
-\end{proposition}
+\end{corollary}
 \begin{proof}
     By \autoref{proposition:lch-sigma-compact}, there exists an exhaustion $\seq{U_n} \subset 2^X$ of $X$ by relatively compact open sets. Denote $U_0 = \emptyset$. For each $n \in \natp$, let $V_n = U_{n+1} \setminus \ol{U_{n-1}}$.
 
-    Let $x \in X$, then there exists $n \in \natp$ such that $x \in U_n \setminus U_{n-1}$. In which case, if $n > 1$, then $x \in U_{n} \setminus \ol{U_{n - 2}} = V_{n-1}$. If $n = 1$, then $x \in U_{2} = V_1$. Thus $\seq{V_n}$ is an open cover of $X$. In addition, for any $m, n \in \natp$ with $m \le n$, $V_m \cap V_n \ne \emptyset$ implies that $n - m < 2$, so $\seq{V_n}$ is locally finite. By (2) of \autoref{proposition:lch-paracompact}, $X$ is paracompact.
+    Let $x \in X$, then there exists $n \in \natp$ such that $x \in U_n \setminus U_{n-1}$. In which case, if $n > 1$, then $x \in U_{n} \setminus \ol{U_{n - 2}} = V_{n-1}$. If $n = 1$, then $x \in U_{2} = V_1$. Thus $\seq{V_n}$ is an open cover of $X$. In addition, for any $m, n \in \natp$ with $m \le n$, $V_m \cap V_n \ne \emptyset$ implies that $n - m < 2$, so $\seq{V_n}$ is locally finite. By (2) of \autoref{theorem:lch-paracompact}, $X$ is paracompact.
 \end{proof}