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

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 $\fU$.