Oopsies daisies.

src/topology/main/c0.tex

@@ -17,7 +17,7 @@ For details regarding the complex-valued cased, in particular its properties as
     \begin{enumerate}
         \item $C_0(X; E) \subset BC(X; E)$.
         \item $C_0(X; E)$ is a closed subspace of $BC(X; E)$ with respect to the uniform topology. In particular, if $E$ is complete, then so is $C_0(X; E)$.
-        \item If $X$ is a LCH space, then $C_c(X; E)$ is a dense subspace of $C_0(X; E)$ with respect to the uniform topology.
+        \item If $X$ is an LCH space, then $C_c(X; E)$ is a dense subspace of $C_0(X; E)$ with respect to the uniform topology.
     \end{enumerate}
 \end{proposition}
 \begin{proof}
@@ -44,7 +44,7 @@ For details regarding the complex-valued cased, in particular its properties as
 
 \begin{proposition}
 \label{proposition:c0-tensor}
-    Let $X$ be a LCH space and $E$ be a locally convex space over $K \in \RC$. Identify $C_0(X; K) \otimes E$ as a subspace of $C_0(X; E)$ under the natural map
+    Let $X$ be an LCH space and $E$ be a locally convex space over $K \in \RC$. Identify $C_0(X; K) \otimes E$ as a subspace of $C_0(X; E)$ under the natural map
     \[
     C_0(X; K) \otimes E \to C_0(X; E) \quad \sum_{j = 1}^n \phi_j \otimes x_j \mapsto \sum_{j = 1}^n x_j \cdot \phi_j
     \]

src/topology/main/compactify.tex

@@ -73,7 +73,7 @@
 
 \begin{lemma}
 \label{lemma:lch-compactification-open}
-    Let $X$ be a LCH space and $(Y, \varphi)$ be a compactification of $X$, then $\varphi(X) \subset Y$ is open. 
+    Let $X$ be an LCH space and $(Y, \varphi)$ be a compactification of $X$, then $\varphi(X) \subset Y$ is open. 
 \end{lemma}
 \begin{proof}
     For each $x \in X$, let $U \in \cn_X(x)$ be a compact neighbourhood. Since $Y$ is a compact Hausdorff space, $\varphi(U)$ is closed by \autoref{proposition:compact-closed}. As $\varphi \in C(X; Y)$ is an embedding, there exists $V \in \cn_Y(\varphi(x))$ such that $\varphi(U) = \varphi(X) \cap V$. Given that $\varphi(X)$ is dense in $Y$, $\varphi(U) = \ol{\varphi(X) \cap V} \supset V$. Therefore $\varphi(U) \in \cn_{Y}(\varphi(x))$, and $\varphi(X)$ is open in $Y$. 
@@ -83,7 +83,7 @@
 
 \begin{definition}[One-Point Compactification]
 \label{definition:alexandroff-compactification}
-    Let $(X, \mathcal{T})$ be a LCH space, then there exists a pair $(X^*, \iota)$ such that:
+    Let $(X, \mathcal{T})$ be an LCH space, then there exists a pair $(X^*, \iota)$ such that:
     \begin{enumerate}
         \item $(X^*, \iota)$ is a compactification of $X$. 
         \item[(U)] For any pair $(Y, \varphi)$ satisfying (1), there exists a unique $\varphi^* \in C(Y; X^*)$ such that the following diagram commutes:

src/topology/main/lch.tex

@@ -20,7 +20,7 @@
 
 \begin{lemma}
 \label{lemma:lch-compact-neighbour}
-    Let $X$ be a LCH space, $K \subset X$ be compact, and $U \in \cn(K)$, then there exits $V \in \cn^o(K)$ relatively compact such that $K \subset V \subset \ol{V} \subset U$.
+    Let $X$ be an LCH space, $K \subset X$ be compact, and $U \in \cn(K)$, then there exits $V \in \cn^o(K)$ relatively compact such that $K \subset V \subset \ol{V} \subset U$.
 \end{lemma}
 \begin{proof}
     For each $x \in K$, there exists $V_x \in \cn^o(x)$ be relatively compact such that $x \in V_x \subset \overline{V_x} \subset U$ by (3) of \autoref{definition:lch}. Since $K$ is compact, there exists $\seqf{x_j} \subset K$ such that
@@ -38,7 +38,7 @@
 
 \begin{lemma}[Urysohn's Lemma (LCH)]
 \label{lemma:lch-urysohn}
-    Let $X$ be a LCH space, $K \subset X$ be compact, and $U \in \cn(K)$, then there exists $F \in C_c(X; [0, 1])$ such that $\supp{F} \subset U$.
+    Let $X$ be an LCH space, $K \subset X$ be compact, and $U \in \cn(K)$, then there exists $F \in C_c(X; [0, 1])$ such that $\supp{F} \subset U$.
 \end{lemma}
 \begin{proof}[Proof, {{\cite[Lemma 4.32]{Folland}}}. ]
     By \autoref{lemma:lch-compact-neighbour}, there exists $V, W \in \cn^o(K)$ relatively compact such that
@@ -61,7 +61,7 @@
 
 \begin{theorem}[Tietze Extension Theorem (LCH)]
 \label{theorem:lch-tietze}
-    Let $X$ be a LCH space, $K \subset X$ be compact, $U \in \cn^o(K)$, and $f \in C(K; \real)$, then there exists $F \in C_c(U; \real)$ such that $F|_K = f$.
+    Let $X$ be an LCH space, $K \subset X$ be compact, $U \in \cn^o(K)$, and $f \in C(K; \real)$, then there exists $F \in C_c(U; \real)$ such that $F|_K = f$.
 \end{theorem}
 \begin{proof}
     By \autoref{lemma:lch-compact-neighbour}, there exists $V, W \in \cn^o(K)$ relatively compact such that $K \subset V \subset \ol{V} \subset U$. 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}.
@@ -79,7 +79,7 @@
 
 \begin{proposition}
 \label{proposition:lch-compactly-generated}
-    Let $X$ be a LCH space, then:
+    Let $X$ be an LCH space, then:
     \begin{enumerate}
         \item $X$ is compactly generated. 
         \item For any uniform space $Y$, $C(X; Y) \subset Y^X$ is closed with respect to the compact-open topology. 
@@ -94,7 +94,7 @@
 
 \begin{proposition}
 \label{proposition:lch-product}
-    Let $\seqi{X}$ be a family of LCH spaces. If $X_i$ is compact for all but finitely many $i \in I$, then $X = \prod_{i \in I}X_i$ is a LCH space. 
+    Let $\seqi{X}$ be a family of LCH spaces. If $X_i$ is compact for all but finitely many $i \in I$, then $X = \prod_{i \in I}X_i$ is an LCH space. 
 \end{proposition}
 \begin{proof}
     By \autoref{proposition:product-hausdorff}, $\prod_{i \in I}X_i$ is Hausdorff. Let $x \in \prod_{i \in I}X_i$ and $i \in I$. If $X_i$ is not compact, let $U_i \in \cn_{X_i}(\pi_i(x))$ be compact. Otherwise, let $U_i = X_i$. Let $U = \prod_{i \in I}U_i$, then since $U_i \ne X_i$ for only finitely many $i \in I$, $U \in \cn_X(x)$. By \hyperref[Tychonoff's Theorem]{theorem:tychonoff}, $U$ is compact. Therefore $X$ is locally compact. 
@@ -108,7 +108,7 @@
 
 \begin{proposition}
 \label{proposition:lch-sigma-compact}
-    Let $X$ be a LCH space, then the following are equivalent:
+    Let $X$ be an LCH space, then the following are equivalent:
     \begin{enumerate}
         \item $X$ is $\sigma$-compact.
         \item There exists an exhaustion of $X$ by compact sets.
@@ -134,7 +134,7 @@
 
 \begin{proposition}
 \label{proposition:lch-partition-of-unity}
-    Let $X$ be a LCH space, $K \subset X$ be compact, and $\seqi{U}$ be an open cover of $K$, then there exists a $C_c$ partition of unity on $K$ subordinate to $\seqi{U}$.
+    Let $X$ be an LCH space, $K \subset X$ be compact, and $\seqi{U}$ be an open cover of $K$, then there exists a $C_c$ partition of unity on $K$ subordinate to $\seqi{U}$.
 \end{proposition}
 \begin{proof}[Proof, {{\cite[Proposition 4.41]{Folland}}}. ]
     Since $K$ is compact, assume without loss of generality that $\seqi{U} = \seqf{U_j}$.
@@ -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 $E \in \ce$, $G_E \subset \ol{G_E} \subset E \subset \ol{E} \subset F_E$.
+    Let $X$ be an 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
@@ -210,7 +210,7 @@
 
 \begin{theorem}
 \label{theorem:lch-paracompact}
-    Let $X$ be a LCH space, then the following are equivalent:
+    Let $X$ be an LCH space, then the following are equivalent:
     \begin{enumerate}
         \item $X$ is paracompact.
         \item There exists a locally finite relatively compact open cover $\cf$ of $X$.

src/topology/main/semicontinuity.tex

@@ -64,7 +64,7 @@
 
 \begin{proposition}
 \label{proposition:semicontinuous-lch}
-    Let $X$ be a LCH space and $f: X \to [0, \infty]$ be lower semicontinuous, then
+    Let $X$ be an LCH space and $f: X \to [0, \infty]$ be lower semicontinuous, then
     \[
     f = \sup_{\substack{\phi \in C_c(X) \\ 0 \le \phi \le f}}\phi
     \]

src/topology/uniform/equicontinuous.tex

@@ -57,7 +57,7 @@
         \item $\cf$ is a relatively compact subset of $C(X; Y)$ with respect to the uniform structure of compact convergence. 
     \end{enumerate}
 
-    Conversely, if $X$ is a LCH space, then (C3) implies (E1) + (E2).
+    Conversely, if $X$ is an LCH space, then (C3) implies (E1) + (E2).
 \end{theorem}
 \begin{proof}
     (E1) $\Rightarrow$ (C1): It is sufficient to show that (ii) is finer than (iii).
@@ -80,7 +80,7 @@
 
     (E1) + (E2) $\Rightarrow$ (C3): Using (C2), assume without loss of generality that $\cf$ is closed in $Y^X$ with respect to the product topology. In which case, $\cf$ is a closed subset of $\prod_{x \in X}\ol{\cf(x)}$ with respect to the product topology. By \hyperref[Tychonoff's Theorem]{theorem:tychonoff} and \autoref{proposition:compact-extensions}, $\cf$ is compact in the product topology. By (C1), $\cf$ is also compact in the compact uniform topology.
 
-    (C3) $\Rightarrow$ (E1): Assume that $X$ is a LCH space. Let $x \in X$ and $U \in \fU$ be symmetric, then there exists a compact neighbourhood $V \in \cn_X(x)$. Since $\cf$ is totally bounded, there exists $\seqf{f_j} \subset \cf$ such that for each $g \in \cf$, there exists $1 \le j \le n$ such that $(f_j \times g)(V) \subset U$. For each $1 \le j \le n$, $f_j \in C(X; Y)$, so there exists $V_j \in \cn_X(x)$ with $V_j \subset V$ such that for any $y \in V_j$, $(f_j(x), f_j(y)) \in U$. Let $W = \bigcap_{j = 1}^n V_j$, then for any $g \in \cf$ with $(f_j \times g)(V) \subset U$ and $y \in W$,  
+    (C3) $\Rightarrow$ (E1): Assume that $X$ is an LCH space. Let $x \in X$ and $U \in \fU$ be symmetric, then there exists a compact neighbourhood $V \in \cn_X(x)$. Since $\cf$ is totally bounded, there exists $\seqf{f_j} \subset \cf$ such that for each $g \in \cf$, there exists $1 \le j \le n$ such that $(f_j \times g)(V) \subset U$. For each $1 \le j \le n$, $f_j \in C(X; Y)$, so there exists $V_j \in \cn_X(x)$ with $V_j \subset V$ such that for any $y \in V_j$, $(f_j(x), f_j(y)) \in U$. Let $W = \bigcap_{j = 1}^n V_j$, then for any $g \in \cf$ with $(f_j \times g)(V) \subset U$ and $y \in W$,  
     \[
     (g(x), f_j(x)), (f_j(x), f_j(y)), (f_j(y), g(y)) \in U \circ U \circ U
     \]