Incremental update.

src/topology/uniform/metric.tex

@@ -22,7 +22,7 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa
     \end{enumerate}
 \end{lemma}
 \begin{proof}
-    (1) $\Rightarrow$ (2): Let $r > 0$, then there exists $V \in \fU$ symmetric such that for any $(x, x'), (y, y') \in V$, $\abs{d(x, y) - d(x', y')} < r$. In particular, for any $(x, y) \in V$, $(x, x), (x, y) \in V$. Thus $d(x, y) < d(x, x) + r = r$. Therefore $V \subset U_r$, and $U_r \in \fU$.
+    (1) $\Rightarrow$ (2): Let $r > 0$, then there exists $V \in \fU$ symmetric such that for any $(x, x'), (y, y') \in V$, $\abs{d(x, y) - d(x', y')} < r$. In particular, for any $(x, y) \in V$, $(x, x), (x, y) \in V$. Thus $d(x, y) < d(x, x) + r = r$, $V \subset U_r$, and $U_r \in \fU$.
 
     (2) $\Rightarrow$ (1): Let $r > 0$, then for any $(x, x'), (y, y') \in U_{r/2}$, $\abs{d(x, y) - d(x', y')} < r$ by the triangle inequality.
 \end{proof}
@@ -229,14 +229,14 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa
     \]
     so $d$ induces the uniformity on $\fU$.
 
-    (3) $\Rightarrow$ (1): By (1) of \ref{definition:pseudometric-uniformity},
+    (3) $\Rightarrow$ (1): By (1) of \ref{definition:pseudometric-uniformity}, if
+    \[
+    U_{n, r} = \bracs{(x, y) \in X \times X| d_n(x, y) < r}
+    \]
+    then
     \[
     \fB = \bracs{\bigcap_{j \in J}U_{j, r} \bigg | J \subset \nat \text{ finite}, r > 0}
     \]
-    where
-    \[
-    U_{j, r} = \bracs{(x, y) \in X \times X| d_n(x, y) < r}
-    \]
     is a fundamental system of entourages for $\fU$. Since for any $r > 0$, there exists $q \in \rational \cap (0, r)$,
     \[
     \bracs{\bigcap_{j \in J}U_{j, r} \bigg | J \subset \nat \text{ finite}, r \in \rational, r > 0}