Polished A-A and added new lines for broken enumerates.

src/cat/cat/cat-func.tex

@@ -13,7 +13,8 @@
         \item[(CAT1)] For any $A, B, A', B' \in \obj{\catc}$, $\mor{A, B}$ and $\mor{A', B'}$ are disjoint or equal, where $\mor{A, B} = \mor{A', B'}$ if and only if $A = A'$ and $B = B'$.
         \item[(CAT2)] For any $A \in \obj{\catc}$, there exists $\text{Id}_A \in \mor{A, A}$ such that $f \circ \text{Id}_A = f$ and $\text{Id}_A \circ g = g$ for all $B, C \in \obj{\catc}$, $f \in \mor{A, B}$, and $g \in \mor{C, A}$.
         \item[(CAT3)] For any $A, B, C, D \in \obj{\catc}$, $f \in \mor{A, B}$, $g \in \mor{B, C}$, and $h \in \mor{C, D}$, $(h \circ g) \circ f = h \circ (g \circ f)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The elements of $\obj{\catc}$ are the \textbf{objects} of $\catc$, and elements of $\mor{A, B}$ are the \textbf{morphisms/arrows} from $A$ to $B$.
 \end{definition}
 

src/cat/cat/tensor.tex

@@ -32,7 +32,8 @@
         A_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & 
         }
         \]
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     The module $A = \bigoplus_{i \in I}A_i$ is the \textbf{direct sum} of $\seqi{A}$. 
 \end{definition}
@@ -204,7 +205,8 @@
         \[
         (x_1, \cdots, \alpha x_j, \cdots, x_n) - \alpha(x_1, \cdots, x_n)
         \]
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     (1), (2): Let $\bigotimes_{j = 1}^n E_j = M/N$ and
     \[

src/cat/cat/universal.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item \textbf{universally attracting} if for every $A \in \obj{\catc}$, there exists a unique $f \in \mor{A, P}$.
         \item \textbf{universally repelling} if for every $A \in \obj{\catc}$, there exists a unique $f \in \mor{P, A}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If $P$ is universally attracting or repelling, then $P$ is a \textbf{universal object}.
 
     If $P, Q \in \obj{\catc}$ are both universally attracting/repelling, then they are isomorphic.
@@ -62,12 +63,14 @@
     \begin{enumerate}
         \item For any $i \in I$, $i \lesssim i$.
         \item For any $i, j, k \in I$ such that $i \lesssim j$ and $j \lesssim k$, $i \lesssim k$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     and one of the following holds:
     \begin{enumerate}
         \item[(3U)] For any $i, j \in I$, there exists $k \in I$ with $i, j \lesssim k$.
         \item[(3D)] For any $i, j \in I$, there exists $k \in I$ with $k \lesssim i, j$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The directed set is \textbf{upward-directed} if it satisfies (3U), and \textbf{downward-directed} if it satisfies (3D).
 \end{definition}
 
@@ -85,7 +88,8 @@
         \item For each $i \in I$, $f^i_i = \text{Id}_{A_i}$.
         \item For each $i, j \in I$ with $i \lesssim j$, $f^i_j \in \mor{A_i, A_j}$.
         \item For each $i, j, k \in I$ with $i \lesssim j \lesssim k$, $f^j_k \circ f^i_j = f^i_k$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If $I$ is upward/downward-directed, then $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ is upward/downward-directed.
 \end{definition}
 

src/cat/gluing/index.tex

@@ -8,7 +8,8 @@
     \begin{enumerate}
         \item[(a)] $\bigcup_{i \in I}U_i = X$.
         \item[(b)] For each $i, j \in I$, either $U_i \cap U_j = \emptyset$, or $f_i|_{U_i \cap U_j} = f_j|_{U_i \cap U_j}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a unique $f: X \to Y$ such that $f|_{U_i} = f_i$ for all $i \in I$.
 \end{lemma}
 \begin{proof}
@@ -16,7 +17,8 @@
     \begin{enumerate}
         \item By assumption (a), $\bracs{x|(x, y) \in \Gamma} = \bigcup_{i \in I}U_i = X$.
         \item For any $x \in X$, there exists $y \in Y$ with $(x, y) \in \Gamma$, and $i \in I$ such that $(x, y) \in \Gamma_i$. If $(x, y') \in \Gamma_j \subset \Gamma$, then $x \in U_i \cap U_j \ne \emptyset$. By assumption (b), $y = y'$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Thus $\Gamma$ is the graph of a function $f: X \to Y$ with $f|_{U_i} = f_i$ for all $i \in I$.
 \end{proof}
 
@@ -27,7 +29,8 @@
         \item[(a)] $\bigcup_{V \in \fF}V = E$.
         \item[(b)] For each $V, W \in \fF$, $T_V|_{V \cap W} = T_W|_{V \cap W}$.
         \item[(c)] $\fF$ is upward-directed with respect to includion.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a unique $T \in \hom(E; F)$ such that $T|_{V} = T_V$ for all $V \in \fF$.
 \end{lemma}
 \begin{proof}

src/cat/tricks/dyadic.tex

@@ -42,7 +42,8 @@
     \begin{enumerate}
         \item[(a)] for each $n \in \natp$, $g_{n+1} + g_{n+1} \le g_n$.
         \item[(b)] For each $x, y \in G$, $x + y \ge x, y$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     For each $x \in \mathbb{D} \cap [0, 1)$, let $\phi(x) = \sum_{n \in M(x)}g_n$, then
     \begin{enumerate}

src/dg/derivative/higher.tex

@@ -9,7 +9,8 @@
     \begin{enumerate}
         \item There exists $V \in \cn_E(x_0)$ such that $f$ is $(n-1)$-fold differentiable on $V$.
         \item The derivative $D_\sigma^{n-1}f: U \to B^{n-1}_\sigma(E; F)$ is derivative at $x_0$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In which case, $D_\sigma(D_\sigma^{n-1}f)(x_0) \in L(E; B^{n-1}_\sigma(E; F))$ is the \textbf{$n$-fold $\sigma$-derivative of $f$ at $x_0$}.
 
     The mapping $f: U \to F$ is \textbf{$n$-fold $\sigma$-differentiable on $U$} if it is $n$-fold $\sigma$-differentiable at every point in $U$. Under the identification $B_\sigma(E; B^{n-1}_\sigma(E; F)) = B_\sigma^{n}(E; F)$ given by \autoref{proposition:multilinear-identify},

src/dg/derivative/mvt.tex

@@ -17,7 +17,8 @@
     \begin{enumerate}
         \item[(a)] $f, g$ are right-differentiable on $[a, b] \setminus N$.
         \item[(b)] For every $x \in [a, b] \setminus N$, $D^+f(x) \le D^+g(x)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then for any $x \in [a, b]$, $f(x) - f(a) \le g(x) - g(a)$.
 \end{lemma}
 \begin{proof}
@@ -51,7 +52,8 @@
         \item $f, g$ are right-differentiable on $[a, b] \setminus N$.
         \item For each $x \in [a, b] \setminus N$, $D^+f(x) \in D^+g(x)B$.
         \item $g$ is non-decreasing.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then
     \[
     f(b) - f(a) \in [g(b) - g(a)]B

src/dg/derivative/sets.tex

@@ -8,7 +8,8 @@
         \item For each $A \in \sigma$, $r(th)/t^n \to 0$ uniformly on $A$.
         \item If $r_t(x) = r(tx)/t^n$, then $r_t \to 0$ as $t \to 0$ with respect to the $\sigma$-uniform topology on $F^E$.
         \item For each $A \in \sigma$, $\seq{a_k} \subset A$, and $\seq{t_k} \subset K \setminus \bracs{0}$ with $t_k \to 0$ as $n \to \infty$, $r(t_ka_k)/t_k^n \to 0$ as $n \to \infty$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $r$ is \textbf{$\sigma$-small of order $n$}.
 
     The set $\mathcal{R}_\sigma^n(E; F)$ is the $K$-vector space of all $\sigma$-small functions of order $n$ from $E$ to $F$. For simplicity, $\mathcal{R}_\sigma(E; F)$ denotes $\mathcal{R}_\sigma^1(E; F)$.
@@ -52,7 +53,8 @@
     \begin{enumerate}
         \item[(a)] For any $r \in \mathcal{R}_\sigma(E; F)$ and $T \in L(F; G)$, $T \circ r \in \mathcal{R}_\sigma(E; G)$.
         \item[(b)] For any $r \in \mathcal{R}_\sigma(E; F)$, $T \in L(E; F)$, and $s \in \mathcal{R}_\tau(F; G)$, $s \circ (T + r) \in \mathcal{R}_\sigma(E; G)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then for any $U \subset E$ and $V \subset F$ open, $f: U \to V$ $\sigma$-differentiable at $x_0 \in U$, $g: V \to F$ $\tau$-differentiable at $f(x_0) \in V$, $g \circ f: U \to F$ is $\sigma$-differentiable at $x_0$ with
     \[
     D_\sigma(g \circ f)(x_0) = D_\tau g(f(x_0)) \circ D_\sigma f(x_0)
@@ -84,12 +86,14 @@
     \begin{enumerate}
         \item Compact sets.
         \item Bounded sets.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then
     \begin{enumerate}
         \item For any $r \in \mathcal{R}_\sigma(E; F)$ and $T \in L(F; G)$, $T \circ r \in \mathcal{R}_\sigma(E; G)$.
         \item For any $r \in \mathcal{R}_\sigma(E; F)$, $T \in L(E; F)$, and $s \in \mathcal{R}_\tau(F; G)$, $s \circ (T + r) \in \mathcal{R}_\sigma(E; G)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     and by \autoref{proposition:chain-rule-sets}, $\sigma$-derivatives and $\tau$-derivatives satisfy the Chain rule.
 \end{proposition}
 \begin{proof}

src/fa/duality/polar.tex

@@ -27,7 +27,8 @@
     \begin{enumerate}
         \item $\emptyset^\circ = \emptyset^\square = F$ and $F^\circ = F^\square = \bracs{0}$. 
         \item For any $A, B \subset E$ and $\lambda \ne 0$, if $\lambda A \subset B$, then $B^\circ \subset \lambda^{-1}A^\circ$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 

src/fa/lc/bornologic.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item For any $U \subset E$ convex and balanced, if $U$ absorbs every bounded set of $E$, then $U \in \cn_E(0)$.
         \item For any seminorm $\rho: E \to [0, \infty)$ that is bounded on all bounded sets of $E$, $\rho$ is continuous. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     If the above holds, then $E$ is a \textbf{bornological space}.
 \end{definition}

src/fa/lc/convex.tex

@@ -138,7 +138,8 @@
         \item For each $i \in I$, $d_i: E \times E \to [0, \infty)$ defined by $(x, y) \mapsto [x - y]_i$ is a pseudo-metric.
         \item The topology induced by $\seqi{d}$ makes $E$ a topological vector space.
         \item For each $i \in I$, $[\cdot]_i: E \to [0, \infty)$ is continuous.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The topology induced by $\seqi{d}$ is the \textbf{vector space topology induced by} $\seqi{[\cdot]}$. In addition,
     \begin{enumerate}
         \item[(U)] For any family $\bracsn{[\cdot]_j}_{j \in J}$ of continuous seminorms on $E$, the vector space topology induced by $\bracsn{[\cdot]_j}_{j \in J}$ is contained in the vector space topology induced by $\seqi{[\cdot]}$.
@@ -159,7 +160,8 @@
         \item If $A$ is convex, then for any $x, y \in E$, $[x + y]_A \le [x]_A + [y]_A$.
         \item If $A$ is circled, then for any $x \in E$ and $\lambda \in K$, $[\lambda x]_A = \abs{\lambda}[x]_A$.
         \item If $A$ is circled, then $\bracs{\rho < 1} \subseteq A \subseteq \bracs{\rho \le 1} \subseteq \ol A$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In particular,
     \begin{enumerate}[start=4]
         \item If $A$ is convex, then $[\cdot]_A$ is a sublinear functional.
@@ -189,7 +191,8 @@
         \item There exists a fundamental system of neighborhoods at $0$ consisting of convex sets.
         \item There exists a fundamental system of neighbourhoods at $0$ consisting of convex, circled, and radial sets.
         \item There exists a family of seminorms $\seqi{[\cdot]}$ that induces the topology on $E$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $E$ is a \textbf{locally convex} space.
 \end{definition}
 \begin{proof}

src/fa/lc/hahn-banach.tex

@@ -134,7 +134,8 @@
         \begin{enumerate}
             \item $|\phi| \le \rho$.
             \item $\dpb{x, \phi}{E} = \inf_{y \in M}\rho(x + y)$.
-        \end{enumerate}
+        \end\{enumerate\}
+
         \item For any $x \in E$ and continuous seminorm $\rho: E \to [0, \infty)$, there exists $\phi \in E^*$ with $|\phi| \le \rho$ and $\dpb{x, \phi}{E} = \rho(x)$.
         \item If $E$ is separated, then for any $x, y \in E$, there exists $\phi \in E^*$ with $\dpb{x, \phi}{E} \ne \dpb{y, \phi}{E}$.
     \end{enumerate}

src/fa/lc/inductive.tex

@@ -21,7 +21,8 @@
         \]
 
         is a fundamental system of neighbourhoods for $E$ at $0$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     The topology $\topo$ is the \textbf{inductive locally convex topology} on $E$ induced by $\seqi{T}$.
 \end{definition}
 \begin{proof}
@@ -64,7 +65,8 @@
         \]
 
         is a fundamental system of neighbourhoods for $E$ at $0$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     The space $E = \bigoplus_{i \in I}E_i$ is the \textbf{locally convex direct sum} of $\seqi{E}$. 
 \end{definition}
@@ -110,7 +112,8 @@
         \]
 
         is a fundamental system of neighbourhoods for $E$ at $0$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The pair $(E, \bracsn{T^i_E}_{i \in I})$ is the \textbf{inductive limit} of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$.
 \end{definition}
 \begin{proof}
@@ -174,7 +177,8 @@
         \begin{enumerate}
             \item[(a)] $B$ is bounded.
             \item[(b)] There exists $n \in \natp$ such that $B \subset E_n$ is bounded.
-        \end{enumerate}
+        \end\{enumerate\}
+
         \item If $E_n$ is complete for each $n \in \natp$, then $E$ is also complete.
     \end{enumerate}
 \end{proposition}
@@ -190,7 +194,8 @@
         \item For each $k \in \natp$, $U_k \in \cn_{E_{n_k}}(0)$.
         \item For each $k \in \natp$, $U_k = U_{k+1} \cap E_{n_k}$.
         \item For each $k \in \natp$, $n^{-1}x_k \not\in U_k$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $V = \bigcup_{k \in \natp}U_k \in \cn_E(0)$ with $V \cap E_{n_k} = U_k$ for all $k \in \natp$. For any $n \in \natp$, $x_k \not\in nU_k = nV \cap E_{n_k}$. Therefore $B$ is not bounded.
 
     (3), $(b) \Rightarrow (a)$: Let $U \in \cn_E(0)$, then $U \cap E_n \in \cn_{E_n}(0)$, so there exists $\lambda \in K$ with $\lambda (U \cap E_n) \supset B$.

src/fa/lc/quotient.tex

@@ -41,7 +41,8 @@
 
         If $F$ is a TVS over $K$ and $f \in L(E; F)$, then $\td f \in L(E; F)$.
         \item If $\seqi{\rho}$ is a family of seminorms that induces the topology on $E$, then their quotients by $M$ induces the topology on $\td E$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The space $\td E = E/M$ is the \textbf{quotient} of $E$ by $M$.
 \end{definition}
 \begin{proof}

src/fa/lc/spaces-of-linear.tex

@@ -6,7 +6,8 @@
     Let $T$ be a set, $\sigma \subset 2^T$ be an ideal, $E$ be a locally convex space over $K$, and $\cf \subset E^T$ be a subspace such that
     \begin{enumerate}
         \item[(B)] For each $f \in \cf$ and $S \in \sigma$, $f(S) \subset E$ is bounded. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     For each $S \in \sigma$ and continuous seminorm $\rho: E \to [0, \infty)$, let
     \[

src/fa/lc/tensor.tex

@@ -23,7 +23,8 @@
         \]
 
         is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     The space $E \otimes_\pi F$ is the \textbf{projective tensor product} of $E$ and $F$, and the mapping $\iota \in L^2(E, F; E \otimes_\pi F)$ is the \textbf{canonical embedding}.
 
@@ -72,7 +73,8 @@
     and the seminorm $\rho = p \otimes q$ is the \textbf{cross seminorm} of $p$ and $q$. Moreover, 
     \begin{enumerate}
         \item[(5)] If the seminorms $\seqi{p}$ define the topology on $E$, and the seminorms $\seqj{q}$ define the topology on $F$, then the seminorms $\bracsn{p_i \otimes q_j| (i, j) \in I \times J}$ define the topology on $E \otimes_\pi F$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{definition}
 \begin{proof}[Proof {{\cite[III.6.3]{SchaeferWolff}}}. ]
@@ -140,7 +142,8 @@
         \item $\sum_{n \in \natp}|\lambda_n| < \infty$. 
         \item $\limv{n}x_n = 0$ and $\limv{n}y_n = 0$.
         \item $z = \sum_{n = 1}^\infty \lambda_n x_n \otimes y_n$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{theorem}
 \begin{proof}
@@ -162,7 +165,8 @@
         \item $v_N = \sum_{k = 1}^{n_N}\lambda_{N, k}x_{N, k} \otimes y_{N, k}$.
         \item For each $1 \le k \le n_N$, $p_N(x_{N, k}), q_N(x_{N, k}) \le 1/M$.
         \item $\sum_{k = 1}^{n_N}|\lambda_k| \le 2^{-N+2}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     From here, let $\seqf{(x_j, y_j)} \subset X \times Y$ such that $u_1 = \sum_{j = 1}^n x_j \otimes y_j$, then
     \[

src/fa/lp/definition.tex

@@ -107,7 +107,8 @@
     \begin{enumerate}
         \item[(a)] $f_n \to f$ strongly pointwise.
         \item[(b)] There exists $g \in L^p(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     then $f_n \to f$ in $L^p(X; E)$.
 \end{proposition}

src/fa/norm/absolute.tex

@@ -12,7 +12,8 @@
     \begin{enumerate}
         \item $E$ is a Banach space.
         \item For any absolutely convergent series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$, there exists $x \in E$ such that $x = \sum_{n = 1}^\infty x_n$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{lemma}
 \begin{proof}
@@ -44,7 +45,8 @@
         \item If $\sum_{n \in P}x_n = \infty$ but $\sum_{n \in N}x_n > -\infty$, then $\sum_{n = 1}^\infty x_n$ converges to $\infty$ unconditionally.
         \item If $\sum_{n \in N}x_n = -\infty$ but $\sum_{n \in P}x_n < \infty$, then $\sum_{n = 1}^\infty x_n$ converges to $-\infty$ unconditionally. 
         \item If $\sum_{n \in \natp}|x_n| < \infty$, then $\sum_{n = 1}^\infty x_n$ converges unconditionally.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In other words, a series in $\real$ converges unconditionally if and only if its positive parts or its negative parts are finite. 
 \end{theorem}
 \begin{proof}

src/fa/norm/multilinear.tex

@@ -8,7 +8,8 @@
         \item For each $x \in E$, $y \mapsto T(x, y)$ is a continuous linear map from $F$ to $G$.
         \item For each $y \in F$, $x \mapsto T(x, y)$ is a continuous linear map from $E$ to $G$.
         \item $E$ is a Banach space.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $T \in L^2(E, F; G)$.
 \end{proposition}
 \begin{proof}

src/fa/norm/normed.tex

@@ -38,12 +38,14 @@
     \begin{enumerate}
         \item[(a)] $\norm{x}_E \le C\norm{y}_F$.
         \item[(b)] $\norm{y - Tx}_F \le \gamma \norm{y}_F$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then for any $y \in F$, there exists $\seq{x_n} \subset E$ such that:
     \begin{enumerate}
         \item $\sum_{n \in \natp}\norm{x_n}_E \le C\norm{y}_F/(1 - \gamma)$.
         \item $\sum_{n = 1}^\infty Tx_n = y$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In particular, if $E$ is a Banach space, then for every $y \in F$, there exists $x \in E$ such that $\norm{x}_E \le C\norm{y}_F/(1 - \gamma)$ and $Tx = y$.
 \end{theorem}
 \begin{proof}
@@ -73,7 +75,8 @@
     \begin{enumerate}
         \item For every $x \in E$, $\sup_{T \in \mathcal{T}}\norm{Tx}_F < \infty$.
         \item $E$ is a Banach space.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $\sup_{T \in \mathcal{T}}\norm{T}_{L(E; F)} < \infty$.
 \end{theorem}
 \begin{proof}

src/fa/norm/separable.tex

@@ -29,7 +29,8 @@
         \item $\bracs{B(x, r)|x \in E, r > 0}$.
         \item $\bracsn{\ol{B(x, r)}|x \in E, r > 0}$. 
         \item Open sets in $E$ with respect to the weak topology. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}

src/fa/order/lattice.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item[(LO1)] For any $x, y, z \in E$ with $x \le y$, $x + z \le y + z$.
         \item[(LO2)] For any $x, y \in E$ and $\lambda > 0$, $x \le y$ implies that $\lambda x \le \lambda y$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{definition}
 
@@ -17,7 +18,8 @@
     \begin{enumerate}
         \item $\sup(A + B) = \sup(A) + \sup(B)$.
         \item $\sup(A) = -\inf (-A)$
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}
@@ -103,12 +105,14 @@
     \begin{enumerate}
         \item[(3)] $|\lambda x| = |\lambda| \cdot |x|$
         \item[(4)] $|x + y| \le |x| + |y|$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     Finally, for any $x, y \in E$ with $x, y \ge 0$, 
     \begin{enumerate}
         \item[(5)] $[0, x] + [0, y] = [0, x + y]$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}
@@ -175,7 +179,8 @@
     \begin{enumerate}
         \item For any $x \in C$ and $\lambda \in \real$ with $\lambda \ge 0$, $\phi(\lambda x) = \lambda \phi(x)$.
         \item For any $x, y \in C$, $\phi(x + y) = \phi(x) + \phi(y)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     then the mapping
     \[
@@ -218,7 +223,8 @@
         \[
         |\phi|(x) = \sup\bracs{\phi(y)|y \in E, |y| \le x}
         \]
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}

src/fa/rs/bv.tex

@@ -67,7 +67,8 @@
         \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}$ is a seminorm on $BV([a, b]; E)$.
         \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}: E^{[a, b]} \to [0, \infty]$ is lower semicontinuous. In particular, for any $M > 0$, $\bracs{[\cdot]_{\text{var}, \rho} \le M} \subset E^{[a, b]}$ is closed. 
         \item For any $f \in BV([a, b]; E)$ and continuous seminorm $\rho$ on $E$, $\sup_{x \in [a, b]}\rho(f(x)) \le \rho(f(a)) + [f]_{\text{var}, \rho}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If $(E, \norm{\cdot}_E)$ is a normed vector space, then
     \begin{enumerate}
         \item[(5)] $f$ has at most countably many discontinuities.
@@ -87,7 +88,8 @@
     \begin{enumerate}
         \item[(a)] $|E_k| \ge N - k$.
         \item[(b)] $E_k \subset I_k^o$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     for $k = 1$.
 
     Let $k \le N$ and suppose inductively that $E_k, I_k$ have been constructed. Let $x_k \in E_k$, then by (b), there exists $\eps > 0$ such that $[x_k - \eps, x_k + \eps] \subset I_k$ and $|E_k \setminus [x_k - \eps, x_k + \eps]| \ge N - k$. Let $y_k \in [x_k - \eps, x_k + \eps]$ such that $\norm{f(x_k) - f(y_k)} \ge 1/n$, $I_{k + 1} = I_k \setminus [x_k - \eps, x_k + \eps]$, and $E_{k+1} = E_k \setminus [x_k - \eps, x_k + \eps]$, then $I_k$ and $E_k$ satisfies (a) and (b).

src/fa/rs/rs-bv.tex

@@ -35,7 +35,8 @@
     \begin{enumerate}
         \item[(a)] For each continuous seminorm $\rho$ on $E$, $[f_\alpha - f]_{u, \rho} \to 0$.
         \item[(b)] $\lim_{\alpha \in A}\int_a^b f_\alpha dG$ exists.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $f \in RS([a, b], G)$ and $\int_a^b f dG = \lim_{\alpha \in A}\int_a^b f_\alpha dG$. In particular,
     \begin{enumerate}
         \item If $H$ is complete, then condition (b) may be omitted.
@@ -55,11 +56,13 @@
     \begin{enumerate}
         \item $[f - f_\alpha]_E < \eps/(3[G]_{\text{var}, F})$.
         \item $\rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps/3$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Since $f_\alpha \in RS([a, b], G)$, there exists $P_0 \in \scp([a, b])$ such that if $P \ge P_0$,
     \begin{enumerate}
         \item[(3)] $\rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG} < \eps/3$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Thus for any $(P, c) \in \scp_t([a, b])$ with $P \ge P_0$,
     \[
     \rho\paren{S(P, c, f, G) - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps

src/fa/tvs/complete-metric.tex

@@ -21,12 +21,14 @@
     \begin{enumerate}
         \item[(a)] $\eta(y - Tx) \le \gamma \eta(y)$.
         \item[(b)] $\rho(x) \le C \eta(y)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then for any $y \in F$, there exists $\seq{x_n} \subset E$ such that:
     \begin{enumerate}
         \item $\sum_{n \in \natp}\rho(x_n) \le C\eta(y)/(1 - \gamma)$.
         \item $y = \limv{N}\sum_{n = 1}^N Tx_n$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In particular,
     \[
     T\braks{B_E\paren{0, \frac{Cr}{(1 - \gamma)}}} \supset B_F(0, r)
@@ -38,12 +40,14 @@
     \begin{enumerate}
         \item[(I)] $\sum_{n = 1}^N\rho(x_n) \le C\eta(y)\sum_{n = 0}^{N-1}\gamma^{n}$.
         \item[(II)] $\eta\paren{y - \sum_{n = 1}^N Tx_n} \le \eta(y)\gamma^N$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By assumption, there exists $x_{N+1} \in E$ such that:
     \begin{enumerate}
         \item[(i)] $\eta\paren{y - \sum_{n = 1}^{N+1} Tx_n} \le \gamma \eta\paren{y - \sum_{n = 1}^N Tx_n} \le \gamma^{N+1}$.
         \item[(ii)] $\rho(x_{N+1}) \le C\eta\paren{y - \sum_{n = 1}^N Tx_n} \le C\eta(y)\gamma^N$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Combining (I) and (ii) shows that $\sum_{n = 1}^N \rho(x_n) \le C \eta(y) \sum_{n = 0}^N \gamma^n$. Therefore there exists $\seq{x_n} \subset E$ such that (I) and (II) holds for all $N \in \natp$.
 
     By (I), $\sum_{n \in \natp}\rho(x_n) \le C\eta(y)\sum_{n \in \natz}\gamma^n = C \eta(y)/(1 - \gamma)$. By (II), $\limv{N}\eta\paren{y - \limv{N}\sum_{n = 1}^N Tx_n} = \limv{N}\eta(y)\gamma^N = 0$.
@@ -55,7 +59,8 @@
     \begin{enumerate}
         \item[(a)] For any $r > 0$, there exists $\delta(r) > 0$ such that $\overline{T(B_E(0, r))} \supset B_F(0, \delta(r))$.
         \item[(b)] $E$ is complete.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then for every $s > r$, $T(B_E(0, s)) \supset B_F(0, \delta(r))$.
 \end{proposition}
 \begin{proof}
@@ -65,12 +70,14 @@
         \item[(ii)] $s_1 = r$.
         \item[(iii)] For all $n \in \natp$, $\overline{T(B_E(0, s_n))} \supset B_F(0, \delta_n)$.
         \item[(iv)] $\rho_1 = \rho$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Let $y_0 \in B(0, r)$ and $x_0 = 0$. Let $N \in \natp$ and suppose inductively that $\bracs{x_n}_1^N \subset E$ has been constructed such that:
     \begin{enumerate}
         \item[(I)] For each $0 \le n \le N - 1$, $\rho(x_{n+1} - x_n) < s_n$.
         \item[(II)] For each $0 \le n \le N$, $\eta(Tx_n - y) \le \rho_{n+1}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By density of $T(x_N + B_E(0, s_N))$ in $Tx_N + B_F(0, \rho_N)$, there exists $x_{N+1} \in T(x_N + B_E(0, s_N))$ such that $\eta(Tx_{N+1} - y) \le \rho_{N+2}$.
 
     By (I), $\seq{x_N}$ is a Cauchy sequence, so
@@ -88,7 +95,8 @@
     \begin{enumerate}
         \item[(a)] For any $r > 0$, there exists $C \ge 0$ such that for any $y \in T(E)$, there exits $x \in T^{-1}(y)$ with $\rho(x) \le C\eta(y)$.
         \item[(b)] $E$ is complete.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $T(E)$ is closed.
 \end{proposition}
 \begin{proof}

src/fa/tvs/completion.tex

@@ -8,11 +8,13 @@
         \item $\wh E$ is a complete separated TVS.
         \item $\iota \in L(E; \wh E)$.
         \item[(U)] For any $(F, T)$ satisfying (1) and (2), there exists a unique $\ol{T} \in L(\wh E; F)$ such that the following diagram commutes:
-    \end{enumerate}
+    \end\{enumerate\}
+
     Moreover,
     \begin{enumerate}
         \item[(4)] $\iota(E)$ is dense in $\wh E$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The pair $(\wh E, \iota)$ is the \textbf{Hausdorff completion} of $E$.
 \end{definition}
 \begin{proof}

src/fa/tvs/continuous.tex

@@ -9,7 +9,8 @@
         \item $T \in UC(E; F)$.
         \item $T \in C(E; F)$.
         \item $T$ is continuous at $0$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $T$ is a \textbf{continuous linear map}. The set $L(E; F)$ denotes the vector space of all continuous linear maps from $E$ to $F$.
 \end{definition}
 \begin{proof}
@@ -48,7 +49,8 @@
         }
         \]
 
-    \end{enumerate}
+    \end\{enumerate\}
+
     The uniformity $\fU$ and its induced topology are the \textbf{product uniformity/topology}, and $E$ equipped with $\fU$ is the \textbf{product TVS} of $\seqi{E}$.
 \end{definition}
 

src/fa/tvs/definition.tex

@@ -8,7 +8,8 @@
     \begin{enumerate}
         \item[(TVS1)] $E \times E \to E$ with $(x, y) \mapsto x + y$ is continuous.
         \item[(TVS2)] $K \times E \to E$ with $(\lambda, x) \mapsto \lambda x$ is continuous.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then the pair $(E, \topo)$ is a \textbf{topological vector space}.
 \end{definition}
 
@@ -50,7 +51,8 @@
     \begin{enumerate}
         \item There exists a unique translation-invariant uniformity $\fU$ on $E$ that induces the topology on $E$.
         \item For each neighbourhood $V \in \cn(0)$, let $U_V = \bracs{(x, y) \in E^2| x - y \in V}$, then for any fundamental system of neighbourhoods $\fB_0$ at $0$, $\fB = \bracs{U_V| V \in \fB_0}$ is a fundamental system of entourages for $\fU$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The space $E$ will always be assumed to be equipped with its translation-invariant uniformity.
 \end{proposition}
 \begin{proof}
@@ -162,12 +164,14 @@
     \begin{enumerate}
         \item[(TVB1)] For each $U \in \fB$, there exists $V \in \fB$ such that $V + V \subset U$.
         \item[(TVB2)] For each $U \in \fB$, $U$ is circled and radial.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Conversely, if $\fB \subset 2^E$ is a family of sets that contain $0$ and satisfies (TVB1) and (TVB2), then there exists a unique topology $\topo$ on $E$ such that:
     \begin{enumerate}
         \item $\topo$ is translation-invariant.
         \item $\fB$ is a fundamental system of neighbourhoods at $0$ for $\topo$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Moreover,
     \begin{enumerate}
         \item[(3)] $(E, \topo)$ is a TVS.
@@ -186,7 +190,8 @@
         \item[(FB1)] For any $V, V' \in \fB$, there exists $W \in \fB$ with $W \subset V \cap V'$. In which case, $U_{V} \cap U_{V'} \supset U_W \in \mathfrak{V}$.
         \item[(UB1)] For any $x \in E$ and $V \in \fB$, $x - x = 0 \in V$, so $\Delta \subset U_V$.
         \item[(UB2)] For any $V \in \fB$, by (TVB1), there exists $W \in \fB$ such that $W + W \subset V$. In which case, for any $x, y, z \in E$ with $x - y, y - z \in W$, $x - z \in V$. Therefore $U_W \circ U_W \subset U_V$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By \autoref{proposition:fundamental-entourage-criterion}, there exists a unique uniformity $\fU$ on $E$ for which $\mathfrak{V}$ is a fundamental system of entourages.
 
     (1): Since $\mathfrak{V}$ is translation-invariant, so is $\fU$.

src/fa/tvs/dual.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item $u \in \hom(E; \real)$ when $E$ is viewed as a vector space over $\real$.
         \item For any $x \in E$, $\dpb{x, \phi}{E} = \dpb{x, u}{E} - i \dpb{ix, u}{E}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Conversely, if $u \in \hom(E; \real)$ and $\phi \in \hom(E; \complex)$ is defined by $\dpb{x, \phi}{E} = \dpb{x, u}{E} - i \dpb{ix, u}{E}$ for all $x \in E$, then $f \in \hom(E; \complex)$.
 \end{proposition}
 \begin{proof}[Proof {{\cite[Proposition 5.5]{Folland}}}. ]

src/fa/tvs/inductive.tex

@@ -9,7 +9,8 @@
         \item For each $i \in I$, $T_i \in L(E_i; E)$.
         \item[(U)] For any topology $\mathcal{S}$ on $E$ satisfying (1) and (2), $\mathcal{S} \subset T$.
         \item For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T_i \in L(E_i; F)$ for all $i \in I$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The topology $\topo$ is the \textbf{inductive topology} on $E$ induced by $\seqi{T}$.
 \end{definition}
 \begin{proof}
@@ -61,7 +62,8 @@
 
         for all $i \in I$.
         \item For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T^i_E \in L(E_i; F)$ for all $i \in I$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The pair $(E, \bracsn{T^i_E}_{i \in I})$ is the \textbf{inductive limit} of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$.
 \end{definition}
 \begin{proof}

src/fa/tvs/metric.tex

@@ -82,7 +82,8 @@
     \begin{enumerate}
         \item[(a)] For each $n \in \natp$, $U_n$ is circled, radial, and contains $0$.
         \item[(b)] For each $n \in \natp$, $U_{n+1} + U_{n+1} \subset U_n$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a pseudonorm $\rho: E \to [0, \infty)$ such that for each $n \in \natp$,
     \[
     U_{n+1} \subset \rho^{-1}([0, 2^{-n})) \subset U_{n}
@@ -110,7 +111,8 @@
 
         so $\rho(\lambda x) \le \rho(x)$.
         \item[(PN3)] Let $x, y \in X$ and $M, N \subset \natp$ finite such that $x \in U_M$ and $y \in U_N$. Assume without loss of generality that $\rho_M + \rho_N < 1$, then there exists a unique $P \subset \nat$ finite such that $\rho_P = \rho_M + \rho_N$. In which case, $U_P \supset U_M + U_N$ by assumption (b). Therefore $\rho(x + y) \le \rho(x) + \rho(y)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     For any $x \in U_{n+1}$, $\rho(x) \le 2^{-n+1} < 2^n$, so $U_{n+1} \subset \rho^{-1}([0, 2^{-n}))$ by \autoref{proposition:dyadic-semigroup-order}. On the other hand, for any $x \in E$ with $\rho(x) < 2^{-n}$, $x \in U_{2^{-n}} = U_n$. This allows showing the remaining seminorm axioms by considering neighbourhoods of the form $\bracs{U_n|n \in \natp}$.
     \begin{enumerate}
         \item[(PN4)] Let $x \in X$ and $n \in \natp$. By assumption (a), there exists $\alpha > 0$ such that for any $\lambda \in K$ with $\abs{\lambda} \ge \alpha$, $x \in \lambda U_n$. Therefore for any $\lambda \in K$ with $\abs{\lambda} \le \alpha^{-1}$, $\lambda x \in U_n$, and $\rho(x) \le 2^{-n}$.

src/fa/tvs/projective.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item For each $i \in I$, $T_i \in L(E; F_i)$.
         \item[(U)] If $\mathfrak{V}$ is a uniformity on $E$ satisfying $(1)$, then $\mathfrak{V} \supset \fU$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Moreover,
     \begin{enumerate}
         \item[(3)] $\fU$ is translation-invariant.
@@ -19,7 +20,8 @@
         \]
 
         is a fundamental system of neighbourhoods for $E$ at $0$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     The uniformity $\fU$ and its topology are the \textbf{projective uniformity/topology} induced by $\seqi{T}$.
 \end{definition}
@@ -74,7 +76,8 @@
 
         for all $i \in I$.
         \item For any TVS $F$ over $K$ and $S \in \hom(F; E)$, $S \in L(F; E)$ if and only if $T^E_i \circ S \in L(F; E_i)$ for all $i \in I$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The pair $(E, \bracsn{T^E_i}_{i \in I})$ is the \textbf{projective limit} of $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$.
 \end{definition}
 \begin{proof}

src/fa/tvs/quotient.tex

@@ -17,7 +17,8 @@
         \]
 
         If $F$ is a TVS over $K$ and $f \in L(E; F)$, then $\td f \in L(E; F)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The space $\td E = E/M$ is the \textbf{quotient} of $E$ by $M$.
 \end{definition}
 \begin{proof}
@@ -32,7 +33,8 @@
     \begin{enumerate}
         \item[(TVB1)] Let $U \in \cn(0)$ be circled and radial. For any $\lambda \in K$ with $\abs{\lambda} \le 1$, $\lambda \pi(U) = \pi(\lambda U) \subset \pi(U)$, so $\pi(U)$ is also circled. For any $x + M \in E/M$, there exists $\lambda \in K$ such that $x \in \lambda U$. In which case, $x \in \lambda U + M = \pi(U)$, so $\pi(U)$ is also radial.
         \item[(TVB2)] For any $U \in \cn(0)$ circled and radial, by \autoref{proposition:tvs-good-neighbourhood-base}, there exists $W \in \cn(0)$ such that $W + W \subset U$. In which case, $\pi(W) + \pi(W) \subset \pi(U)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By \autoref{proposition:tvs-0-neighbourhood-base}, there exists a unique translation-invariant topology on $E/M$ such that $\fB$ is a fundamental system of neighbourhoods at $0$, which must be the quotient topology on $E/M$. In which case, the quotient topology is a vector space topology by (3) of \autoref{proposition:tvs-0-neighbourhood-base}.
 
     (2), (3), (U): By \autoref{definition:quotient-topology}.

src/fa/tvs/spaces-of-linear.tex

@@ -115,7 +115,8 @@
         \]
 
         is an isomorphism.
-    \end{enumerate}
+    \end\{enumerate\}
+
     which allows the identification
     \[
     \underbrace{B_{\sigma}(E; B_{\sigma}(E; \cdots)))}_{k \text{ times}} = B^k_{\sigma}(E; F)
@@ -166,7 +167,8 @@
     \begin{enumerate}
         \item[(a)] There exists a dense subset $S \subset E$ such that $T_\alpha x \to Tx$ strongly for all $x \in S$.
         \item[(b)] $\bracs{T_\alpha|\alpha \in A}$ is uniformly equicontinuous. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     then $T_\alpha \to T$ in $L_s(E; F)$.
 \end{proposition}

src/measure/bochner-integral/bochner.tex

@@ -52,7 +52,8 @@
     \begin{enumerate}
         \item[(a)] $f_n \to f$ strongly pointwise.
         \item[(b)] There exists $g \in L^1(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     then $\int f d\mu = \limv{n}\int f_n d\mu$. 
 \end{theorem}
@@ -76,7 +77,8 @@
     \begin{enumerate}
         \item For any $x \in E$ and $A \in \cm$, $I_\lambda(x \cdot \one_A) = x \mu(A)$.
         \item For any $f \in L^1(X, |\mu|; E)$, $\normn{I_\lambda f}_{G} \le \norm{\lambda}_{L^2(E, F; G)} \cdot \norm{f}_{L^1(X, |\mu|; E)}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     For any $f \in L^1(X; E)$, $I_\lambda f = \int \lambda(f, d\mu)$ is the \textbf{Bochner integral} of $f$ with respect to $\mu$ and $\lambda$. 
 \end{definition}
@@ -96,7 +98,8 @@
     \begin{enumerate}
         \item[(a)] $f_n \to f$ strongly pointwise.
         \item[(b)] There exists $g \in L^1(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     then $\int \lambda(f , d\mu) = \limv{n}\int \lambda(f_n, d\mu)$. 
 \end{theorem}

src/measure/bochner-integral/strongly.tex

@@ -11,7 +11,8 @@
         \begin{enumerate}
             \item[(a)] For each $n \in \natp$, $\norm{f_n}_E \le \norm{f}_E$.
             \item[(b)] $\norm{f_n(x) - f(x)}_E \to 0$ pointwise as $n \to \infty$.
-        \end{enumerate}
+        \end\{enumerate\}
+
     \end{enumerate}
 
     If the above holds, then $f$ is a \textbf{strongly measurable} function. 
@@ -39,7 +40,8 @@
     \begin{enumerate}
         \item For any strongly measurable functions $f, g: X \to E$ and $\lambda \in K$, $\lambda f + g$ is strongly measurable. 
         \item For any strongly measurable functions $\bracs{f_n: X \to E|n \in \natp}$ and $f: X \to E$, if $f_n \to f$ strongly pointwise, then $f$ is strongly measurable. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}

src/measure/lebesgue-integral/complex.tex

@@ -104,7 +104,8 @@
     \begin{enumerate}
         \item $f_n \to f$ pointwise.
         \item There exists $g \in \mathcal{L}^1(X)$ such that $\abs{f_n} \le \abs{g}$ for all $n \in \natp$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $\int fd\mu = \limv{n}\int f_n d\mu$.
 \end{theorem}
 \begin{proof}[Proof {{\cite[Theorem 2.24]{Folland}}}. ]

src/measure/measurable-maps/metric.tex

@@ -76,7 +76,8 @@
         \item[(a)] For each $y \in Y$, $y \in \ol{N(y)^o}$.
         \item[(b)] $\bigcap_{y \in Y}N(y) \ne \emptyset$.
         \item[(c)] For any $y_0 \in Y$, $\bracs{y \in Y|y_0 \in N(y)} \in \cb_Y$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Then, for any $f: X \to Y$, the following are equivalent:
     \begin{enumerate}
         \item $f$ is $(\cm, \cb_Y)$-measurable.
@@ -84,7 +85,8 @@
         \begin{enumerate}
             \item[(i)] For each $x \in X$ and $n \in \natp$, $f_n(x) \in N(f(x))$.
             \item[(ii)] $f_n \to f$ pointwise.
-        \end{enumerate}
+        \end\{enumerate\}
+
         \item There exists a sequence $\seq{f_n}$ of $(\cm, \cb_Y)$-measurable simple functions such that $f_n \to f$ pointwise.
     \end{enumerate}
 \end{proposition}
@@ -146,6 +148,7 @@
         \bracs{y \in E|y_0 \in N(y)} = \bracs{y \in E|\norm{y_0}_E \le \norm{y}_E} \in \cb_E
         \]
 
-    \end{enumerate}
+    \end\{enumerate\}
+
     By \autoref{proposition:measurable-simple-separable}, (1) and (2) are equivalent.
 \end{proof}

src/measure/measurable-maps/real-valued.tex

@@ -24,7 +24,8 @@
         \item $G = \limsup_{n \to \infty}f_n$.
         \item $g = \limsup_{n \to \infty}f_n$.
         \item $\limv{n}f_n$ (if it exists).
-    \end{enumerate}
+    \end\{enumerate\}
+
     In addition, if the above functions are $\real$-valued, then they are $(\cm, \cb_{\real})$-measurable.
 \end{proposition}
 \begin{proof}

src/measure/measure/complete.tex

@@ -20,7 +20,8 @@
         \item There exists an extension $\ol{\mu}$ of $\mu$ as a measure on $\cm$.
         \item $(X, \ol{\cm}, \ol{\mu})$ is a complete measure space.
         \item[(U)] For any complete measure space $(X, \cf, \nu)$ where $\cf \supset \cm$ and $\nu|_\cm = \mu$, $\cf \supset \ol{\cm}$ and $\nu|_{\ol{\cm}} = \ol{\mu}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     and space $(X, \ol{\cm}, \mu)$ is the \textbf{completion} of $(X, \cm, \mu)$.
 \end{definition}
 \begin{proof}

src/measure/measure/kolmogorov.tex

@@ -16,13 +16,15 @@
         \item[(a)] Every finite measure on $\prod_{j = 1}^n X_j$ is regular.
         \item[(b)] $X_n$ is Hausdorff.
         \item[(c)] $X_n$ is separable.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Let $\bracs{\mu_{I}| I \subset \natp \text{ finite}}$ be consistent Borel probability measures, then for any $\seq{B_n}$ where:
     \begin{enumerate}
         \item[(d)] For each $n \in \nat$, $B_n \in \cb_{\prod_{j = 1}^n X_j}$.
         \item[(e)] For each $n \in \nat$, $B_{n+1} \subset B_n \times X_{n+1}$.
         \item[(f)] There exists $\eps > 0$ such that $\mu_{[n]}(B_n) > \eps$ for all $n \in \natp$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Then there exists $\seq{K_n}$ such that for every $n \in \natp$,
     \begin{enumerate}
         \item $K_n \subset \prod_{j = 1}^n X_j$ is compact.
@@ -52,7 +54,8 @@
         \item[(a)] Every finite measure on $\prod_{j \in J} X_j$ is regular.
         \item[(b)] $X_j$ is Hausdorff.
         \item[(c)] $X_j$ is separable.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Let $\bracs{\mu_{I}| I \subset \natp \text{ finite}}$ be consistent Borel probability measures, then there exists a unique probability measure $\mu: \bigotimes_{i \in I}\cb_{X_i} \to [0, 1]$ such that for any $J \subset I$ finite, $\mu = \mu_J \circ \pi_J^{-1}$.
 \end{theorem}
 \begin{proof}[Proof {{\cite[Theorem 1.14]{Baudoin}}}. ]
@@ -71,7 +74,8 @@
         \item $K_n \subset B_n$.
         \item $K_{n+1} \subset K_n \times X_{n+1}$.
         \item $\mu(K_n) \ge \eps/2$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Let $N \in \natp$ and $x \in \prod_{j = 1}^N X_j$ such that $x \in \bigcap_{n \ge N}\pi_{[N]}(K_n)$. By compactness and (4), there exists $x_{N+1} \in X_{N+1}$ such that $(x_1, \cdots, x_N, x_{N+1}) \in \bigcap_{n > N}\pi_{[N+1]}(K_n)$. Thus there exists $x \in \prod_{i \in I}X_i$ such that $x \in \pi_{[n]}^{-1}(K_n)$ for all $n \in \natp$, and
     \[
     x \in \bigcap_{n \in \natp}\pi_{[n]}^{-1}(K_n) \subset \bigcap_{n \in \natp}\pi_{[n]}^{-1}(B_n) \ne \emptyset

src/measure/measure/lebesgue-stieltjes.tex

@@ -44,7 +44,8 @@
         \]
 
         \item[(U)] For any function $G: \real \to \real$ satisfying (1), $F - G$ is constant.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Conversely, if $F: \real \to \real$ is a Stieltjes function, then there exists a unique Borel measure $\mu_F: \cb_\real \to [0, \infty]$ satisfying (1), and $\mu_F$ is the \textbf{Lebesgue-Stieltjes measure} associated with $F$.
 \end{definition}
 \begin{proof}[Proof {{\cite[Theorem 1.16]{Folland}}}. ]

src/measure/measure/measure.tex

@@ -14,13 +14,15 @@
     \begin{enumerate}
         \item[(M1)] $\mu(\emptyset) = 0$.
         \item[(M2)] For any $\seq{E_n} \subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{n \in \natp}E_n} = \sum_{n \in \natp} \mu(E_n)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In which case, $(X, \cm, \mu)$ is a \textbf{measure space}.
 
     If $\mu: \cm \to [0, \infty]$ instead satisfies (M1) and
     \begin{enumerate}
         \item[(M2')] For any $\seqf{E_j} \subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{j = 1}^n E_j} = \sum_{j = 1}^n \mu(E_j)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $\mu$ is a \textbf{finitely-additive measure}.
 \end{definition}
 
@@ -100,7 +102,8 @@
         \item[(a)] $\sigma(\mathcal{P}) = \cm$.
         \item[(b)] $\mu(E) = \nu(E)$ for all $E \in \mathcal{P}$.
         \item[(c)] There exists $\seq{E_n} \subset \mathcal{P}$ such that $E_n \upto X$ and $\mu(E_n) < \infty$ for all $n \in \natp$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $\mu = \nu$.
 \end{theorem}
 \begin{proof}
@@ -122,7 +125,8 @@
         \]
 
         by continuity from below (\autoref{proposition:measure-properties}).
-    \end{enumerate}
+    \end\{enumerate\}
+
     so $\alg(F)$ is a $\lambda$-system. By (a) and Dynkin's $\pi$-$\lambda$ theorem (\autoref{theorem:pi-lambda}), $\alg(F) = \cm$.
 
     Let $\seq{E_n}$ as in assumption (c), then $\mu(E_n \cap F) = \mu(E_n \cap F)$ for all $n \in \natp$ and $F \in \cm$. Thus by continuity from below (\autoref{proposition:measure-properties}),
@@ -163,7 +167,8 @@
         \[
         \int g df_*\mu = \int g \circ f d\mu
         \]
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{definition}
 \begin{proof}

src/measure/measure/outer.tex

@@ -115,7 +115,8 @@
             \item[(a)] $\nu|_{\cm \cap \cn} \le \mu_{\cm \cap \cn}$.
             \item[(b)] For any $E \in \cm \cap \cn$ with $\mu(E) < \infty$, $\mu(E) = \nu(E)$.
             \item[(c)] If $\mu$ is $\sigma$-finite, then $\nu = \mu$.
-        \end{enumerate}
+        \end\{enumerate\}
+
     \end{enumerate}
 \end{theorem}
 \begin{proof}[Proof {{\cite[Theorem 1.14]{Folland}}}. ]

src/measure/measure/product.tex

@@ -8,7 +8,8 @@
     \begin{enumerate}
         \item For each $E \in \cm$ and $F \in \cn$, $\mu \otimes \nu(E \times F) = \mu(E)\nu(F)$. 
         \item[(U)] For any measure $\lambda: \cm \otimes \cn \to [0, \infty]$, $\lambda \le \mu$. For any $A \in \cm \otimes \cn$ with $\mu(A) < \infty$, $\lambda(A) = \mu(A)$. In particular, if $\mu$ is $\sigma$-finite, then $\lambda = \mu$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     The measure $\mu \otimes \nu$ is the \textbf{product} of $\mu$ and $\nu$. 
 \end{definition}
@@ -57,7 +58,8 @@
     \begin{enumerate}
         \item For any $E \in \cm \otimes \cn$, $x \in X$, and $y \in Y$, $\bracs{z \in Y|(x, z) \in E} \in \cm$ and $\bracs{z \in X|(z, y) \in E} \in \cn$.
         \item For any measure space $(Z, \cf)$, $(\cm \otimes \cn, \cf)$-measurable function $f: X \times Y \to Z$, $x \in X$, and $y \in Y$, $f(x, \cdot)$ is $(\cn, \cf)$-measurable and $f(\cdot, y)$ is $(\cm, \cf)$-measurable. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{lemma}
 \begin{proof}
@@ -100,8 +102,10 @@
         \int_{X \times Y}f(z)\mu \otimes \nu(dz) &= \int_X\int_Y f(x, y)\nu(dy)\mu(dx) \\
         &= \int_{Y}\int_X f(x, y)\mu(dx)\nu(dy)
         \end{align*}
-        \end{enumerate}
+        \end\{enumerate\}
-    \end{enumerate}
+
+    \end\{enumerate\}
+
     
 \end{theorem}
 \begin{proof}

src/measure/measure/regular.tex

@@ -31,6 +31,7 @@
         \item[(a)] $X$ is a LCH space.
         \item[(b)] Every open set of $X$ is $\sigma$-compact.
         \item[(c)] For any $K \subset X$ compact, $\mu(K) < \infty$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $\mu$ is a regular measure.
 \end{theorem}

src/measure/measure/semifinite.tex

@@ -11,7 +11,8 @@
         \mu(E) = \sup\bracs{\mu(F)| F \in \cm, F \subset E, \mu(F) < \infty}
         \]
 
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $\mu$ is a \textbf{semifinite measure}.
 \end{definition}
 \begin{proof}

src/measure/measure/sigma-finite.tex

@@ -7,6 +7,7 @@
     \begin{enumerate}
         \item There exists $\seq{E_n} \subset \cm$ pairwise disjoint such that $\bigsqcup_{n \in \nat}E_n = X$ and $\mu(E_n) < \infty$ for all $n \in \nat$.
         \item There exists $\seq{E_n} \subset \cm$ such that $E_n \upto X$ and $\mu(E_n) < \infty$ for all $n \in \nat$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $\mu$ is a \textbf{$\sigma$-finite measure}.
 \end{definition}

src/measure/radon/c0.tex

@@ -8,7 +8,8 @@
         \item $I = I^+ - I^-$.
         \item $I^+ \perp I^-$. 
         \item $\norm{I^+}_{C_0(X; \real)^*}, \norm{I^-}_{C_0(X; \real)^*} \le \norm{I}_{C_0(X; \real)^*}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{lemma}
 \begin{proof}

src/measure/radon/radon.tex

@@ -115,7 +115,8 @@
     \begin{enumerate}
         \item[(a)] For any $U \subset X$ open, $U$ is $\sigma$-compact.
         \item[(b)] For any $K \subset X$ compact, $\mu(K) < \infty$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $\mu$ is a regular measure on $X$.
 \end{proposition}
 \begin{proof}

src/measure/radon/riesz.tex

@@ -71,7 +71,8 @@
         \]
 
         As this holds for all $\eps > 0$, $\mu^*(E) \le \sum_{n \in \natp}\mu^*(E_n)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Therefore $\mu^*: 2^E \to [0, \infty]$ is an outer measure.
 
     To see that all Borel sets are $\mu^*$-measurable, let $E \subset X$ and $U \in \topo$. First suppose that $E$ is open. Let $f \prec E \cap U$, then for any $g \prec E \setminus \supp{f}$, $\supp{f} \cap \supp{g} = \emptyset$ and $f + g \prec E$, so

src/measure/sets/algebra.tex

@@ -38,7 +38,8 @@
     \begin{enumerate}
         \item For any $A, B \in \alg$, $A \cap B \in \alg$.
         \item For any $A, B \in \alg$, $A \setminus B \in \alg$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If $\alg$ is a $\sigma$-algebra, then:
     \begin{enumerate}
         \item[(1')] For any $\seq{A_n} \in \alg$, $\bigcap_{n \in \natp}A_n \in \alg$.

src/measure/sets/borel.tex

@@ -98,7 +98,8 @@
         \item Open sets of $X$.
         \item $\bracs{B(x, r)|x \in X, r > 0}$.
         \item $\bracsn{\ol{B(x, r)}|x \in X, r > 0}$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}

src/measure/sets/elementary.tex

@@ -8,7 +8,8 @@
         \item[(P1)] $\emptyset \in \ce$.
         \item[(P2)] For any $A, B \in \ce$, $A \cap B \in \ce$.
         \item[(E)] For any $E, F \in \ce$ with $E \subset F$, there exists $\seqf{E_j} \subset \ce$ such that $E \setminus F = \bigsqcup_{j = 1}^n E_j$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If $X \in \ce$, then (E) may be replaced with
     \begin{enumerate}
         \item[(E')] For any $E \in \ce$, there exists $\seqf{E_j} \subset \ce$ such that $E^c = \bigsqcup_{j = 1}^n E_j$.

src/measure/sets/lambda.tex

@@ -60,7 +60,8 @@
         \braks{\bigcup_{n \in \natp}E_n} \cap F = \bigcup_{n \in \natp}E_n \cap F \in \lambda(\mathcal{P})
         \]
 
-    \end{enumerate}
+    \end\{enumerate\}
+
     so $\cm(\ce)$ is a $\lambda$-system.
 
     Since $\mathcal{P}$ is a $\pi$-system, $\cm(\mathcal{P}) \supset \mathcal{P}$, so $\cm(\mathcal{P}) = \lambda(\mathcal{P})$. Thus for any $E \in \lambda(\mathcal{P})$ and $F \in \lambda(\mathcal{P})$, $E \cap F \in \lambda(\mathcal{P})$. Therefore $\cm(\lambda(\mathcal{P})) \supset \mathcal{P}$, $\cm(\lambda(\mathcal{P})) = \lambda(\mathcal{P})$, and $\lambda(\mathcal{P})$ satisfies (P2). By \autoref{lemma:pi-lambda}, $\lambda(\mathcal{P})$ is a $\sigma$-algebra.

src/measure/vector/complex.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item[(M1)] $\mu(\emptyset) = 0$.
         \item[(M2)] For any $\seq{E_n} \subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{n \in \natp}E_n} = \sum_{n \in \natp} \mu(E_n)$ where the sum converges absolutely. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     By \hyperref[Riemann's Rearrangement Theorem]{theorem:riemann-rearrangement}, (M2) implies that $\mu$ can only take at most one value in $\bracs{-\infty, \infty}$. 
 \end{definition}
 
@@ -101,7 +102,8 @@
     \begin{enumerate}
         \item[(i)] For each $1 \le k \le n$, $M_k = m(A_k) - \mu(A_k) > 0$.
         \item[(ii)] For each $1 \le k \le n - 1$, $A_{k+1} \subset A_k$ with $\mu(A_{k+1}) - \mu(A_k) > M_{k}/2$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     Let $A_n \in \cm$ with $A_{n+1} \subset A_n$ such that $\mu(A_{n+1}) - \mu(A_n) > M_n/2$. Since $A_{n+1} \cap P = \emptyset$ and $\mu(P) = M$, $A_{n+1}$ cannot be positive, so
     \[
@@ -157,7 +159,8 @@
         \item $\mu^+ \perp \mu^-$.
         \item $\mu = \mu^+ - \mu^-$.
         \item[(U)] For any other pair $(\nu^+, \nu^-)$ satisfying (1) and (2), $\mu^+ = \nu^+$ and $\mu^- = \nu^-$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{theorem}
 \begin{proof}

src/measure/vector/variation.tex

@@ -96,7 +96,8 @@
     \begin{enumerate}
         \item $M(X, \cm; E)$ equipped with $\norm{\cdot}_{\text{var}}$ is a normed vector space over $K$.
         \item If $E$ is a Banach space, then so is $M(X, \cm; E)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{definition}
 \begin{proof}

src/measure/vector/vector.tex

@@ -7,7 +7,8 @@
         \begin{enumerate}
         \item[(M1)] $\mu(\emptyset) = 0$.
         \item[(M2)] For any $\seq{A_n} \subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{n \in \natp}A_n} = \sum_{n \in \natp} \mu(A_n)$ where the sum converges absolutely. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{definition}
 

src/process/markov/definition.tex

@@ -31,7 +31,8 @@
     \begin{enumerate}
         \item For each $t \ge 0$, $\bp_t \one = \one$.
         \item For each $t \ge 0$, $\bp_t$ is a positive linear functional.
-    \end{enumerate}
+    \end\{enumerate\}
+
     known as the \textbf{semigroup} of $\bracs{P_t|t \ge 0}$.
 \end{definition}
 \begin{proof}

src/topology/functions/set-systems.tex

@@ -36,7 +36,8 @@
         \item $\mathfrak{E}(\sigma, \fU)$ generates a uniformity $\fV$ on $X^T$.
         \item The topology induced by $\fV$ is finer than the $\sigma$-open topology on $T^X$.
         \item If $\mathfrak{E}(\sigma, \fU)$ forms a fundamental system of entourages for $\fV$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The uniformity $\fV$ is the \textbf{$\sigma$-uniformity}, and the topology induced by $\fV$ is the \textbf{topology of uniform convergence on the sets $\sigma$}/\textbf{$\sigma$-uniform topology} on $X^T$.
 \end{definition}
 \begin{proof}
@@ -57,7 +58,8 @@
         E(S, V) \circ E(S, V) \subset E(S, V \circ V) \subset E(S, U)
         \]
 
-    \end{enumerate}
+    \end\{enumerate\}
+
     By \autoref{proposition:fundamental-entourage-criterion}, $\mathfrak{E}$ is a fundamental system of entourages for the uniformity that it generates.
 \end{proof}
 
@@ -111,7 +113,8 @@
         \item The product topology on $X^T$.
         \item The $\sigma$-open topology, where $\sigma$ is the collection of all finite sets.
         \item (If $X$ is a uniform space) The $\mathfrak{F}$-uniform topology, where $\fF = \bracs{F| F \subset X \text{ finite}}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     This topology is the \textbf{topology of pointwise convergence} on $X^T$.
 \end{definition}
 \begin{proof}

src/topology/functions/uniform.tex

@@ -23,7 +23,8 @@
     \begin{enumerate}
         \item $C(X; Y) \subset Y^X$ is closed with respect to the uniform topology.
         \item If $X$ is a uniform space, then $UC(X; Y) \subset Y^X$ is closed with respect to the uniform topology. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     In particular, if $Y$ is complete, then the above spaces are complete.
 \end{proposition}
 \begin{proof}
@@ -49,3 +50,12 @@
     If $Y$ is complete, then $Y^T$ with the uniform topology is complete by \autoref{proposition:set-uniform-complete}. Thus $C(T; X)$ and $UC(T; X)$ are both complete subspaces by \autoref{proposition:complete-closed}.
 \end{proof}
 
+
+\begin{corollary}
+\label{corollary:uniform-limit-continuous-generated}
+    Let $X$ be a topological space, $\sigma \subset 2^X$ be an ideal such that $X$ is $\sigma$-generated, and $Y$ be a uniform space, then $C(X; Y) \subset Y^X$ is closed with respect to the $\sigma$-uniformity.
+\end{corollary}
+\begin{proof}
+    Let $f \in \overline{C(X; Y)} \subset Y^X$ with respect to the $\sigma$-uniformity. By \autoref{proposition:uniform-limit-continuous}, $f \in C(S; Y)$ for all $S \in \sigma$, so $f \in C(X; Y)$ by (3) of \autoref{definition:final-topology}. 
+\end{proof}
+

src/topology/main/baire.tex

@@ -9,7 +9,8 @@
         \item For any $\seq{A_n} \subset 2^X$ nowhere dense, $\bigcup_{n \in \nat^+}A_n \subsetneq X$.
         \item For any $\seq{A_n} \subset 2^X$ closed with empty interior, $\bigcup_{n \in \nat^+}A_n$ has empty interior.
         \item For any $\seq{U_n} \subset 2^X$ open and dense, $\bigcap_{n \in \nat^+}U_n$ is dense.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $X$ is a \textbf{Baire space}.
 \end{definition}
 \begin{proof}
@@ -28,7 +29,8 @@
     \begin{enumerate}
         \item[(a)] For all $n > 1$, $\ol V_n \subset U_n \cap V_{n - 1} \subset U$.
         \item[(b)] $\bigcap_{j \in \natp} \ol V_j$ is non-empty.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $X$ is a Baire space.
 \end{lemma}
 \begin{proof}

src/topology/main/compact.tex

@@ -9,7 +9,8 @@
         \item For every family $\seqi{E}$ of closed sets with $\bigcap_{j \in J}E_j \ne \emptyset$ for all $J \subset I$ finite, $\bigcap_{i \in I}E_i \ne \emptyset$.
         \item Every filter in $X$ has a cluster point.
         \item Every ultrafilter in $X$ converges.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $X$ is \textbf{compact}.
 \end{definition}
 \begin{proof}
@@ -89,7 +90,8 @@
     \begin{enumerate}
         \item For any $x \in X$ and $U \in \cn_{X \times Y}^o(\bracs{x} \times Y)$, there exists $V \in \cn_X(x)$ such that $V \times Y \subset U$. 
         \item For any $A \subset X$ and $U \in \cn_{X \times Y}^o(A \times Y)$, there exists $V \in \cn_X(A)$ such that $V \times Y \subset U$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{lemma}
 \begin{proof}

src/topology/main/connected.tex

@@ -8,7 +8,8 @@
         \item For any $\emptyset \ne U, V \subset X$ open with $U \cup V = X$, $U \cap V \ne \emptyset$.
         \item There exists no surjective $f \in C(X; \bracs{0, 1})$.
         \item For any $U \subset X$ open and closed, either $U = \emptyset$ or $U = X$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $X$ is \textbf{connected}.
 \end{definition}
 \begin{proof}

src/topology/main/continuity.tex

@@ -8,7 +8,8 @@
     \begin{enumerate}
         \item For each $V \in \cn(f(x))$, $f^{-1}(V) \in \cn(x)$.
         \item For each filter base $\fB \subset 2^X$ converging to $x$, $f(\fB)$ converges to $f(x)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $f$ is \textbf{continuous at} $x \in X$.
 
     The following are also equivalent:
@@ -16,7 +17,8 @@
         \item For each $U \subset Y$ open, $f^{-1}(U)$ is open in $X$.
         \item $f$ is continuous at every $x \in X$.
         \item For each convergent filter base $\fB \subset 2^X$, $f(\fB)$ is convergent.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $f$ is \textbf{continuous}.
 
     The collection $C(X; Y)$ is the space of all continuous functions from $X$ to $Y$.
@@ -39,7 +41,8 @@
     \begin{enumerate}
         \item[(a)] $\bigcup_{i \in I}U_i = X$.
         \item[(b)] For each $i, j \in I$, either $U_i \cap U_j = \emptyset$, or $f_i|_{U_i \cap U_j} = f_j|_{U_i \cap U_j}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a unique $f \in C(X; Y)$ such that $f|_{U_i} = f_i$ for all $i \in I$.
 \end{lemma}
 \begin{proof}

src/topology/main/definition.tex

@@ -9,6 +9,7 @@
         \item[(O2)] For any $U, V \in \topo$, $U \cap V \in \topo$.
         \item[(O3)] For any $\seqi{U} \subset \topo$, $\bigcup_{i \in I}U_i \in \topo$.
     \end{enumerate}
+
     The elements of $\topo$ are known as \textbf{open sets}, and the pair $(X, \topo)$ is known as a \textbf{topological space}.
 \end{definition}
 
@@ -47,6 +48,7 @@
         \item For every $x \in X$, there exists $U \in \cb$ such that $x \in U$.
         \item For every $x \in X$ and $U \subset X$ open with $x \in U$, there exists $V \in \cb$ such that $x \in V \subset U$.
     \end{enumerate}
+
     In which case,
     \[
     \topo = \topo(\cb) = \bracs{\bigcup_{i \in I}U_i \bigg | \seqi{U} \subset \cb, I \text{ index set}}
@@ -58,6 +60,7 @@
         \item[(TB1)] For every $x \in X$, there exists $U \in \cb$ such that $x \in U$.
         \item[(TB2)] For every $x \in X$ and $U, V \in \cb$ such that $x \in U \cap V$, there exists $W \in \cb$ such that $x \in W \subset U \cap V$.
     \end{enumerate}
+
     then $\topo(\cb)$ is a topology on $X$, and $\cb$ is a base for $\topo(\cb)$.
 \end{definition}
 \begin{proof}
@@ -94,10 +97,10 @@
 
 \begin{definition}[Initial Topology]
 \label{definition:initial-topology}
-    Let $X$ be a set, $\bracsn{(Y_j, \topo_i)}$ be a family of topological spaces, and $\seqi{f}$ be a family of maps such that $f_i: X \to Y_i$ for each $i \in I$, then there exists a topology $\topo$ on $X$ such that:
+    Let $X$ be a set, $\bracsn{(Y_i, \topo_i)}_{i \in I}$ be a family of topological spaces, and $\seqi{f}$ be a family of maps such that $f_i: X \to Y_i$ for each $i \in I$, then there exists a topology $\topo$ on $X$ such that:
     \begin{enumerate}
         \item For each $i \in I$, $f_i \in C(X; Y_i)$.
-        \item[(U)] If $\mathcal{S}$ is a topology on $X$ satisfying $(1)$, then $\mathcal{S} \supset \topo$.
+        \item[(U)] If $\mathcal{S}$ is a topology on $X$ satisfying (1), then $\mathcal{S} \supset \topo$.
         \item The family
 
         \[
@@ -107,13 +110,49 @@
 
         is a base for $\topo$.
     \end{enumerate}
-    The topology $\topo$ is known a the \textbf{initial/weak topology} generated by the maps $\seqi{f}$.
+
+    The topology $\topo$ is the \textbf{initial/weak topology} generated by the maps $\seqi{f}$.
 \end{definition}
 \begin{proof}
-    Let $\topo$ be the topology genereated by sets of the form $\ce = \bracs{f_i^{-1}(U_i)| i \in I, U_i \in \topo_i}$. Let $\topo$ be the topology generated by $\ce$, then
+    Let $\topo$ be the topology genereated by sets of the form $\ce = \bracs{f_i^{-1}(U_i)| i \in I, U_i \in \topo_i}$, then
     \begin{enumerate}
         \item For each $i \in I$, $\topo \supset \bracs{f_i^{-1}(U)|U \in \topo_i}$, so $f_i \in C(\topo; Y_i)$.
         \item If $\mathcal{S}$ is a topology such that $f_i \in C(X, \mathcal{S}; Y_i)$, then $\bracs{f_i^{-1}(U)|U \in \topo_i} \subset \mathcal{S}$. Thus $\ce \subset \mathcal{S}$ and $\mathcal{S} \supset \topo$.
         \item By \autoref{definition:generated-topology}, $\cb$ is a base for $\topo$.
     \end{enumerate}
 \end{proof}
+
+\begin{definition}[Final Topology]
+\label{definition:final-topology}
+    Let $X$ be a set, $\bracsn{(Y_i, \topo_i)}_{i \in I}$ be a family of topological spaces, and $\seqi{f}$ be a family of maps such that $f_i: Y_i \to X$ for each $i \in I$, then there exists a topology $\topo$ on $X$ such that:
+    \begin{enumerate}
+        \item For each $i \in I$, $f_i \in C(Y_i; X)$.
+        \item[(U)] If $\mathcal{S}$ is a topology on $X$ satisfying (1), then $\mathcal{S} \subset \topo$.
+        \item For any topological space $Z$ and $F: X \to Z$, $F \in C(X; Z)$ if and only if $F \circ f_i \in C(Y_i; X)$ for all $i \in I$. 
+    \end{enumerate}
+
+    The topology $\topo$ is the \textbf{final topology} generated by the maps $\seqi{f}$.
+\end{definition}
+\begin{proof}
+    Let
+    \[
+    \topo = \bracsn{U \subset X| f_i^{-1}(U) \in \topo_i \forall i \in I}
+    \]
+
+    then since for each $i \in I$, $\topo_i$ is a topology on $Y_i$, $\topo$ is a topology on $X$. 
+
+    (1): By definition, for any $i \in I$ and $U \in \topo$, $f_i^{-1}(U) \in \topo_i$, so $f_i \in C(Y_i; X)$. 
+
+    (U): For any topology $\mathcal{S}$ satisfying (1) and $U \in \mathcal{S}$, $f_i^{-1}(U) \in \mathcal{T}_i$, so $\mathcal{S} \subset \mathcal{T}$. 
+
+    (3): Let $F: X \to Z$ such that $F \circ f_i \in C(Y_i; X)$ for all $i \in I$, then for any $U \subset Z$ open, $f_i^{-1}(F^{-1}(U)) \in \topo_i$ for all $i \in I$. Hence $F^{-1}(U) \in \topo$ and $F \in C(X; Z)$. 
+\end{proof}
+
+\begin{definition}[Generated Topology]
+\label{definition:ideal-generated-topology}
+    Let $X$ be a topological space and $\sigma \subset 2^X$ be an ideal, then $X$ is \textbf{$\sigma$-generated} if the topology of $X$ is the final topology generated by $\bracs{\iota_S: S \to X|S \in \sigma}$. 
+
+    If $\kappa \subset 2^X$ is the collection of precompact sets of $X$, and $X$ is generated by $\kappa$, then $X$ is \textbf{compactly generated}. 
+\end{definition}
+
+

src/topology/main/filters.tex

@@ -25,7 +25,8 @@
     \begin{enumerate}
         \item[(FB1)] For any $E, F \in \fB$, there exists $G \in \fB$ such that $G \subset E \cap F$.
         \item[(FB2)] $\emptyset \not\in \fB$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Conversely, if $\fB \subset 2^X$ is a non-empty collection that satisfies (FB1) and (FB2), then $\fB$ is a base for the filter
     \[
     \fF = \bracs{F \subset X| \exists E \in \fB: E \subset F}
@@ -52,7 +53,8 @@
     \begin{enumerate}
         \item $f(\fB) = \bracs{f(E)| E \in \fB}$ is also a filter base.
         \item If $\fB$ is an ultrafilter base, then $f(\fB)$ is also an ultrafilter base. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}
@@ -117,7 +119,8 @@ The smallest filter $\fF(\fB_0) \subset 2^X$ containing $\fB_0$ is the filter \t
         \item $\fF$ is maximal with respect to inclusion.
         \item For any $E \subset X$, either $E \in \fF$ or $E^c \in \fF$.
         \item For any $\seqf{F_j} \subset X$ such that $\bigcup_{j = 1}^n F_j \in \fF$, there exists $1 \le j \le n$ such that $F_j \in \fF$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $\fF$ is an \textbf{ultrafilter}.
 \end{definition}
 \begin{proof}
@@ -157,7 +160,8 @@ The smallest filter $\fF(\fB_0) \subset 2^X$ containing $\fB_0$ is the filter \t
         \item $\fF \supset \cn(x)$.
         \item For each ultrafilter $\fU \supset \fF$, $\fU \supset \cn(x)$.
         \item There exists a fundamental system of neighbourhoods $\cb(x) \subset \cn(x)$ such that for every $E \in \cb$, there exists $F \in \fB$ with $F \subset E$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $x$ is a \textbf{limit point} of $\fB$, and $\fB$ \textbf{converges} to $x$.
 
     If $A \subset X$ and $\fB \subset 2^A$, then $\fB$ \textbf{converges} to $x$ if $\fF(\fB) \supset \bracsn{U \cap A| U \in \cn(x)}$.
@@ -178,7 +182,8 @@ The smallest filter $\fF(\fB_0) \subset 2^X$ containing $\fB_0$ is the filter \t
         \item $x \in \bigcap_{E \in \fF}\overline{E}$.
         \item There exists a fundamental system of neighbourhoods $\cb(x) \subset \cn(x)$ such that for every $E \in \cb$ and $f \in \fB$, $E \cap F \ne \emptyset$.
         \item There exists a filter $\fU \supset \fB$ that converges to $x$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $x$ is a \textbf{cluster/accumulation point} of $\fB$. In particular, if $\fF$ is an ultra filter, then (6) implies that the limit points and cluster points of $\fF$ coincide.
 \end{definition}
 \begin{proof}

src/topology/main/hausdorff.tex

@@ -12,7 +12,8 @@
         \item Every filter in $X$ converges to at most one point.
         \item For any index set $I$, the diagonal $\Delta$ is closed in $X^I$.
         \item The diagonal $\Delta$ is closed in $X \times X$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $X$ is a \textbf{T2/Hausdorff} space.
 \end{definition}
 \begin{proof}

src/topology/main/lch.tex

@@ -8,7 +8,8 @@
         \item For any $x \in X$, there exists $K \in \cn(x)$ compact.
         \item For any $x \in X$, $\cn(x)$ admits a fundamental system of neighbourhoods consisting of compact sets.
         \item For any $x \in X$, $\cn(x)$ admits a fundamental system of neighbourhoods consisting of precompact sets.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $X$ is a \textbf{locally compact Hausdorff (LCH)} space.
 \end{definition}
 \begin{proof}
@@ -76,6 +77,35 @@
     then by the \hyperref[gluing lemma for continuous functions]{lemma:gluing-continuous}, $\ol F \in C_c(X; \real)$ with $\ol F|_K = F|_K = f$ and $\supp{F} \subset \supp{\eta} \subset V \subset U$.
 \end{proof}
 
+\begin{proposition}
+\label{proposition:lch-compactly-generated}
+    Let $X$ be a 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. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1): Let $U \subset X$ such that $U \cap K$ is open in $K$ for all $K \subset X$ compact. For any $x \in U$, there exists a compact neighbourhood $K \in \cn(x)$. In which case, $U \supset U \cap K \in \cn(x)$, so $U \in \cn(x)$ for all $x \in U$. By \autoref{lemma:openneighbourhood}, $U$ is open. 
+
+    (2): By \autoref{proposition:compact-uniform-open}, the compact-open topology coincides with the compact-uniform topology on $C(X; Y)$. Since $X$ is compactly generated, $C(X; Y) \subset Y^X$ is closed with respect to the compact-open topology by \autoref{corollary:uniform-limit-continuous-generated}. 
+\end{proof}
+
+
+\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. 
+\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. 
+\end{proof}
+
+
+\subsection{Paracompactness and LCH Spaces}
+\label{subsection:lch-paracompact}
+
+
+
 \begin{proposition}[{{\cite[Proposition 4.39]{Folland}}}]
 \label{proposition:lch-sigma-compact}
     Let $X$ be a LCH space, then the following are equivalent:
@@ -92,7 +122,8 @@
         \item[(a)] For each $0 \le k \le n$, $U_k$ is a precompact open set.
         \item[(b)] For each $0 \le k < n$, $\overline{U_k} \subset U_{k+1}$.
         \item[(c)] For each $1 \le k \le n$, $U_k \supset \bigcup_{j = 1}^k K_j$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By \autoref{lemma:lch-compact-neighbour}, there exists $U_{n+1} \in \cn^o(\overline{U_n} \cup K_{n+1})$ precompact. In which case, by (c),
     \[
     U_{n+1} \supset \ol{U_n} \cup K_{n+1} \supset \bigcup_{j = 1}^n K_j \cup K_{n+1} = \bigcup_{j = 1}^{n+1}K_j
@@ -154,7 +185,8 @@
     \begin{enumerate}
         \item[(a)] $\ol{E} \subset \bigcup_{x \in X_E}N_x$.
         \item[(b)] For every $x \in X_E$, $N_x \cap E \ne \emptyset$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Let $X_{\ce} = \bigcup_{E \in \ce}X_\ce$, and for each $E \in \ce$, let
     \[
     G_E = \bigcup_{\substack{x \in X_\ce \\ \ol{N_x} \subset E}}N_x
@@ -238,12 +270,4 @@
     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.
 \end{proof}
 
-\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. 
-\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. 
-\end{proof}
-
 

src/topology/main/neighbourhoods.tex

@@ -36,12 +36,14 @@
         \item[(F2)] For any $A, B \in \cn_\topo(x)$, $A \cap B \in \cn_\topo(x)$.
         \item[(V1)] For every $A \in \cn_\topo(x)$, $x \in A$.
         \item[(V2)] For every $V \in \cn_\topo(x)$, there exists $W \in \cn_\topo(x)$ such that $V \in \cn_\topo(y)$ for all $y \in W$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Conversely, if $\cn: X \to 2^X$ is a mapping such that
     \begin{enumerate}
         \item $\cn(x) \ne \emptyset$ for all $x \in X$.
         \item $\cn(x)$ satisfies (F1), (F2), (V1), and (V2).
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a unique topology $\topo \subset 2^X$ such that $\cn = \cn_\topo$.
 \end{proposition}
 

src/topology/main/normal.tex

@@ -23,7 +23,8 @@
         \begin{enumerate}
             \item[(a)] $U_1 = B^c$.
             \item[(b)] For any $p, q \in \rational \cap [0, 1]$ with $p < q$, $\overline{U_p} \subset U_q$.
-        \end{enumerate}
+        \end\{enumerate\}
+
         \item There exists $f \in C(X; [0, 1])$ with $f|_A = 0$ and $f|_B = 1$.
     \end{enumerate}
 \end{lemma}

src/topology/main/product.tex

@@ -40,7 +40,7 @@
 
 \begin{proposition}
 \label{proposition:productfilterconvergence}
-Let $\bracsn{(X_i, \topo_i)}_{i \in I}$ be a family of topological spaces and $\B$ be a filter base on $\prod_{i \in I}X_i$, then $\fB$ converges to $x \in \prod_{i \in I}X_i$ if and only if $\pi_i(\fB)$ converges to $\pi_i(x)$ for all $i \in I$.
+Let $\bracsn{(X_i, \topo_i)}_{i \in I}$ be a family of topological spaces and $\fB$ be a filter base on $\prod_{i \in I}X_i$, then $\fB$ converges to $x \in \prod_{i \in I}X_i$ if and only if $\pi_i(\fB)$ converges to $\pi_i(x)$ for all $i \in I$.
 \end{proposition}
 \begin{proof}
     $(\Rightarrow)$: Let $i \in I$ and $U \in \cn(\pi_i(x))$, then $\pi_i^{-1}(U) \in \cn(x)$. Since $\fB$ converges to $x$, there exists $B \in \fB$ with $B \subset \pi_i^{-1}(U)$. In which case, $\pi_i(B) \subset U$ and $\pi_i(\fB)$ converges to $\pi_i(x)$.

src/topology/main/quotient.tex

@@ -12,7 +12,8 @@
     \begin{enumerate}
         \item For any $U \subset Y$, $U$ is open if and only if $\pi^{-1}(U)$ is open.
         \item $\pi \in C(X; Y)$, and for any $U \subset X$ saturated and open, $\pi(U)$ is open.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $\pi$ is a \textbf{quotient map}.
 \end{definition}
 \begin{proof}
@@ -37,7 +38,8 @@
         \]
 
         \item $\pi$ is a quotient map.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The space $(\td X, \pi)$ is the \textbf{quotient} of $X$ by $\sim$.
 \end{definition}
 \begin{proof}

src/topology/main/regular.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item For each $x \in X$ and $A \subset X$ closed with $x \not\in A$, there exists $U \in \cn(x)$ and $V \in \cn(A)$ such that $U \cap V = \emptyset$.
         \item For each $x \in X$, the closed neighbourhoods of $x$ forms a fundamental system of neighbourhoods at $x$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If $X$ is a T1 space such that the above holds, then $X$ is \textbf{regular}.
 \end{definition}
 \begin{proof}[Proof {{\cite[Proposition 1.4.11]{Bourbaki}}}. ]

src/topology/metric/metric.tex

@@ -9,7 +9,8 @@
         \item[(M)] For any $x, y \in X$ with $x \ne y$, $d(x, y) > 0$.
         \item[(PM2)] For any $x, y \in X$, $d(x, y) = d(y, x)$.
         \item[(PM3)] For any $x, y, z \in X$, $d(x, z) \le d(x, y) + d(y, z)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The pair $(X, d)$ is a \textbf{metric space}, which comes with the metric uniformity induced by $d$, and the corresponding topology.
 \end{definition}
 

src/topology/uniform/cauchy.tex

@@ -12,7 +12,8 @@
     \begin{enumerate}
         \item $A, B$ are $V$-small.
         \item $A \cap B \ne \emptyset$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $A \cup B$ is $V \circ V$-small.
 \end{lemma}
 \begin{proof}

src/topology/uniform/completion.tex

@@ -19,7 +19,8 @@
 
 
         Moreover, if $f \in UC(X; Y)$, then $F \in UC(\wh X; Y)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Moreover,
     \begin{enumerate}
         \item[(4)] For any symmetric entourage $V \in \fU$, let
@@ -46,14 +47,16 @@
         \item[(FB1)] Let $\wh U, \wh V \in \wh \fB$. By \autoref{lemma:symmetricfundamentalentourage}, there exists a symmetric entourage $W \in \fU$ such that $W \subset U \cap V$. In which case, for any $(\fF, \mathfrak{G}) \in \wh W$, there exists $E \in \fF \cap \mathfrak{G}$ with $E \times E \subset W \subset U \cap V$. Thus $\wh W \subset \wh U \cap \wh V$.
         \item[(UB1)] Let $\wh U \in \wh \fB$ and $\fF \in \wh X$, then since $\fF$ is Cauchy, there exists $E \in \fF$ such that $E \times E \subset U$, so $(\fF, \fF) \in \wh U$.
         \item[(UB2)] Let $\wh U \in \wh \fB$. By \autoref{lemma:symmetricfundamentalentourage}, there exists a symmetric entourage $W \in \fU$ such that $W \circ W \subset U$. For any $\fF, \mathfrak{G}, \mathfrak{H} \in \wh X$ such that $(\fF, \mathfrak{G}), (\mathfrak{G}, \mathfrak{H}) \in \wh W$, there exists $W$-small sets $E \in \fF \cap \mathfrak{G}$ and $F \in \fF \cap \mathfrak{H}$. Since $\mathfrak{G}$ is a filter, $E \cap F \ne \emptyset$ by (F2) and (F3). By \autoref{lemma:small-intersect}, $E \cup H$ is $W \circ W$-small and thus $U$-small. Using (F1), $E \cup H \in \fF \cap \mathfrak{H}$, so $(\fF, \mathfrak{H}) \in \wh U$. Therefore $\wh W \circ \wh W \subset U$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By \autoref{proposition:fundamental-entourage-criterion}, there exists a unique uniformity $\wh \fU \supset \wh \fB$. Moreover, $\wh \fB$ consists of symmetric entourages by construction.
 
     (1, Hausdorff): It is sufficient to show that $\Delta$ is closed and use (6) of \autoref{definition:hausdorff}. Let $(\fF, \mathfrak{G}) \in \ol{\Delta}$, then $(\fF, \mathfrak{G}) \in U$ for all $U \in \fU$ closed. Let $\fB = \bracs{F \cup G| F \in \fF, G \in \mathfrak{G}}$, then
     \begin{enumerate}
         \item[(FB1)] For any $F \cup G, F' \cup G' \in \fB$, $(F \cup G) \cap  (F' \cup G') \supset (F \cap F') \cup (G \cap G') \in \fB$.
         \item[(FB2)] By (F3), $\emptyset \not\in \fF \cup \mathfrak{G}$, so $\emptyset \not\in \fB$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Thus $\fB$ is a filter base by \autoref{proposition:filterbasecriterion}, and the filter $\mathfrak{H}$ generated by $\fB$ is contained in $\fF$ and $\mathfrak{G}$. By \autoref{proposition:goodentourages}, for every $U \in \fU$, there exists a $U$-small set $E \in \fF \cap \mathfrak{G} \subset \fB \subset \mathfrak{H}$. So $\mathfrak{H} \subset \fF, \mathfrak{G}$ is a Cauchy filter. By minimality of $\fF$ and $\mathfrak{G}$, $\fF = \mathfrak{G} = \mathfrak{H}$.
 
     (2): For each $x \in X$, $\cn(x)$ is a minimal Cauchy filter by (1) of \autoref{proposition:cauchyfilterlimit}. Define $\iota: X \to \wh X$ by $x \mapsto \cn(x)$. Let $\wh U \in \wh \fU$ and $(\cn(x), \cn(y)) \in \wh U$, then there exists a $U$-small set $E \in \cn(x) \cap \cn(y)$. By (V1), $(x, y) \in E \times E \in U$.
@@ -104,7 +107,8 @@
         }
         \]
 
-    \end{enumerate}
+    \end\{enumerate\}
+
     known as the \textbf{Hausdorff uniform space associated with} $(X, \fU)$.
 \end{definition}
 \begin{proof}[Proof {{\cite[Proposition 2.8.16]{Bourbaki}}}. ]
@@ -167,7 +171,8 @@
 
 
         Moreover, $\ol{F}(\wh X) = \overline{F(X)}$, and $\ol{F}$ is an embedding.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In particular, by \autoref{proposition:dense-product}, there is a natural isomorphism
     \[
     \prod_{i \in I}\wh X_i \iso \wh{\prod_{i \in I}X_i}

src/topology/uniform/definition.tex

@@ -41,7 +41,8 @@
         \item[(U1)] For every $U \in \fU$, $U \supset \Delta = \bracs{(x, x)| x \in X}$.
         \item[(U2)] For any $U \in \fU$, $U^{-1} \in \fU$.
         \item[(U3)] For any $U \in \fU$, there exists $V \in \fU$ such that $V \circ V \subset U$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The elements of $\fU$ are called the \textbf{entourages} of $\fU$, and the pair $(X, \fU)$ is a \textbf{uniform space}.
 
     For any $x, y \in X$ and $U \in \fU$, $x$ and $y$ are \textbf{$U$-close} if $(x, y) \in U$.
@@ -79,7 +80,8 @@
         \item[(FB1)] For each $U, V \in \fB$, there exists $W \in \fB$ such that $W \subset U \cap V$.
         \item[(UB1)] For each $V \in \fB$, $\Delta \subset V$.
         \item[(UB2)] For each $V \in \fB$, there exists $W \in \fB$ such that $W \circ W \subset V$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a unique uniformity $\fU \subset 2^{X \times X}$, which is given by
     \[
     \fU = \bracs{U \subset X \times X| \exists V \in \fB: V \subset U}
@@ -176,7 +178,8 @@ V = (\bracs{x} \times V)(x) = (U \cup (\bracs{x} \times V))(x) = W(x) \in \cn(x)
     \begin{enumerate}
         \item $V \circ M \circ V \in \cn(M)$.
         \item Let $\fB$ be the set of all symmetric entourages, then $\ol{M} = \bigcap_{V \in \fB}V \circ M \circ V$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     with respect to the product topology on $X \times X$.
 \end{proposition}
 \begin{proof}
@@ -223,7 +226,8 @@ V = (\bracs{x} \times V)(x) = (U \cup (\bracs{x} \times V))(x) = W(x) \in \cn(x)
     \begin{enumerate}
         \item $\mathfrak{O} = \bracs{U^o| U \in \fU}$
         \item $\mathfrak{K} = \bracsn{\overline{U}| U \in \fU}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By \autoref{lemma:symmetricfundamentalentourage}, there exists fundamental systems of entourages for $\fU$ consisting of symmetric and open/closed sets.
 \end{proposition}
 \begin{proof}
@@ -267,7 +271,8 @@ V = (\bracs{x} \times V)(x) = (U \cup (\bracs{x} \times V))(x) = W(x) \in \cn(x)
         \item $X$ is Hausdorff.
         \item $X$ is regular.
         \item $\Delta = \bigcap_{U \in \fU}U$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $X$ is \textbf{separated}.
 \end{definition}
 \begin{proof}

src/topology/uniform/equicontinuous.tex

@@ -66,4 +66,19 @@
 \end{proof}
 
 
+\begin{corollary}
+\label{corollary:arzela-locally-compact}
+    Let $X$ be a LCH space, $Y$ be a uniform space, and $\cf \subset C(X; Y)$ such that:
+    \begin{enumerate}[label=(E\arabic*)]
+        \item $\cf$ is equicontinuous.
+        \item For each $x \in X$, $\cf(x) = \bracs{f(x)|f \in \cf}$ is precompact in $Y$.
+    \end{enumerate}
+
+    then $\cf$ is a precompact subset of $C(X; Y)$ with respect to the compact uniformity. 
+\end{corollary}
+\begin{proof}
+    By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, $\cf$ is a precompact subset of $Y^X$. By \autoref{proposition:lch-compactly-generated}, $C(X; Y)$ is a closed subset of $Y^X$. Therefore $\cf$ is a precompact subset of $C(X; Y)$. 
+\end{proof}
+
+
 

src/topology/uniform/metric.tex

@@ -10,11 +10,13 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa
         \item[(PM1)] For any $x \in X$, $d(x, x) = 0$.
         \item[(PM2)] For any $x, y \in X$, $d(x, y) = d(y, x)$.
         \item[(PM3)] For any $x, y, z \in X$, $d(x, z) \le d(x, y) + d(y, z)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If $d$ satisfies the above and
     \begin{enumerate}
         \item[(M)] For any $x, y \in X$ with $x \ne y$, $d(x, y) > 0$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $d$ is a \textbf{metric}.
 \end{definition}
 
@@ -62,7 +64,8 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa
         \item For each $U \subset X$, $U$ is open if and only if for every $x \in U$, there exists $J \subset I$ finite and $r > 0$ such that $\bigcap_{j \in J}B_j(x, r) \subset U$.
         \item For each $i \in I$, $d_i \in UC(X \times X; [0, \infty))$.
         \item[(U)] For any other uniformity $\mathfrak{V}$ satisfying (4), $\mathfrak{U} \subset \mathfrak{V}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The uniformity $\fU$ is the \textbf{pseudometric uniformity} induced by $\seqi{d}$, and the topology induced by $\fU$ is the \textbf{pseudometric topology} on $X$ induced by $\seqi{d}$.
 \end{definition}
 \begin{proof}
@@ -112,7 +115,8 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa
         \item[(a)] $U_{0} = X \times X$.
         \item[(b)] For each $n \in \natz$, $U_n$ is symmetric.
         \item[(c)] For each $n \in \natz$, $U_{n + 1} \circ U_{n+1} \subset U_n$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a pseudometric $d: X \times X \to [0, 1]$ such that
     \[
     U_{n+1} \subset E(d, 2^{-n}) \subset U_{n-1}
@@ -141,7 +145,8 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa
         \]
 
         As this holds for all such $\seqf{x_j}$ and $\seqf[m]{y_j}$, $d(x, z) \le d(x, y) + d(y, z)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     so $d$ is a pseudometric.
 
     For any $(x, y) \in U_{n+1}$, $d(x, y) \le \rho(x, y) < 2^{-n}$, so $U_{n+1} \subset E(d, 2^{-n})$.
@@ -250,7 +255,8 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa
         \item[(a)] For each $1 \le k \le n$, $V_k$ is symmetric.
         \item[(b)] For each $1 \le k \le n$, $V_k \subset U_k$.
         \item[(c)] For each $1 \le k < n$, $V_{k+1} \circ V_{k+1} \subset V_{k}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Let $W = V_n \cap U_{n+1}$, then by \autoref{lemma:symmetricfundamentalentourage}, there exists $V_{n+1} \in \fU$ symmetric such that $V_{n+1} \circ V_{n+1} \subset W$. Thus $\bracs{V_k|1 \le k \le n + 1} \subset \fU$ satisfies (a), (b), and (c) for $n + 1$.
 
     Let $V_0 = X \times X$, then by \autoref{lemma:uniform-sequence-pseudometric}, there exists a pseudometric $d: X \times X \to [0, \infty)$ such that for each $n \in \natp$,

src/topology/uniform/uc.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item For every $V \in \mathfrak{V}$, there exists $V' \in \fU$ such that $(f(x), f(y)) \in V$ whenever $(x, y) \in V'$.
         \item For every $V \in \mathfrak{V}$, $(f \times f)^{-1}(V) \in \fU$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $f$ is a \textbf{uniformly continuous} function.
 
     The collection $UC(X; Y)$ denotes the set of all uniformly continuous functions from $X$ to $Y$.
@@ -34,7 +35,8 @@
     \begin{enumerate}
         \item For each $i \in I$, $f_i \in UC(X; Y_i)$.
         \item[(U)] If $\mathfrak{V}$ is a uniformity on $X$ satisfying $(1)$, then $\mathfrak{V} \supset \fU$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Moreover,
     \begin{enumerate}
         \item[(3)] The family