Cleanup

src/topology/functions/set-systems.tex

@@ -7,10 +7,12 @@
     \[
     M(S, U) = \bracs{f \in X^T| f(S) \subset U}
     \]
+
     and
     \[
     \ce(\mathfrak{S}, \topo) = \bracs{M(S, U)| S \in \mathfrak{S}, U \in \topo}
     \]
+
     then the topology generated by $\ce$ is the \textbf{$\mathfrak{S}$-open topology} on $T^X$.
 
     If $\cb \subset \topo$ generates $\topo$, then $\ce(\mathfrak{S}, \cb)$ generates the $\mathfrak{S}$-open topology.
@@ -23,10 +25,12 @@
     \[
     E(S, U) = \bracs{(f, g) \in X^T|(f(x), g(x)) \in U \forall x \in S}
     \]
+
     and
     \[
     \mathfrak{E}(\mathfrak{S}, \fU) = \bracs{E(S, U)| S \in \mathfrak{S}, U \in \fU}
     \]
+
     then
     \begin{enumerate}
         \item $\mathfrak{E}(\mathfrak{S}, \fU)$ generates a uniformity $\fV$ on $X^T$.
@@ -46,13 +50,15 @@
         \[
         E(T, U \cap U') \subset E(S \cup S', U \cap U') \subset E(S, U) \cap E(S', U')
         \]
+
         \item[(UB1)] For any $U \in \fU$, $\Delta \subset U$. Thus the diagonal in $X^T$ is in $E(S, U)$ for any $S \in \mathfrak{S}$.
         \item[(UB2)] For any $U \in \fU$, there exists $V \in \fV$ with $V \circ V \subset U$. Thus for any $S \in \mathfrak{S}$,
         \[
         E(S, V) \circ E(S, V) \subset E(S, V \circ V) \subset E(S, U)
         \]
+
     \end{enumerate}
-    By \ref{proposition:fundamental-entourage-criterion}, $\mathfrak{E}$ is a fundamental system of entourages for the uniformity that it generates.
+    By \autoref{proposition:fundamental-entourage-criterion}, $\mathfrak{E}$ is a fundamental system of entourages for the uniformity that it generates.
 \end{proof}
 
 \begin{proposition}
@@ -61,6 +67,7 @@
     \[
     d_{i, S}: X^T \times X^T \quad (f, g) \mapsto \sup_{x \in S}d_i(f(x), g(x))
     \]
+
     then
     \begin{enumerate}
         \item $\bracs{d_{i, S}| i \in I, S \in \mathfrak{S}}$ is a family of pseudometrics induces the $\mathfrak{S}$-uniformity on $X^T$.
@@ -68,6 +75,7 @@
         \[
         \bracs{\bigcap_{j \in J}E(d_{j, S}, r)|J \subset I \text{ finite}, r > 0, S \in \mathfrak{S}}
         \]
+
         is a fundamental system of entourages for the $\mathfrak{S}$-uniformity on $X^T$.
     \end{enumerate}
 \end{proposition}
@@ -76,18 +84,21 @@
     \[
     \bigcap_{j \in J}E(d_{j, S}, r) \subset E\paren{S, \bigcap_{j \in J}E(d_j, r)} \subset E(S, U)
     \]
+
     and the uniformity induced by $\bracs{d_{i, S}| i \in I, S \in \mathfrak{S}}$ contains the $\mathfrak{S}$-uniformity.
 
-    On the other hand, for any $i \in I$ and $r > 0$, $E(d_j, r/2) \in \fU$ by \ref{definition:pseudometric-uniformity}. Therefore $E(S, E(d_j, r/2)) \subset E(d_{j, S}, r)$, so the $\mathfrak{S}$-uniformity contains the induced uniformity.
+    On the other hand, for any $i \in I$ and $r > 0$, $E(d_j, r/2) \in \fU$ by \autoref{definition:pseudometric-uniformity}. Therefore $E(S, E(d_j, r/2)) \subset E(d_{j, S}, r)$, so the $\mathfrak{S}$-uniformity contains the induced uniformity.
 
-    (2): If $\mathfrak{S}$ is upward-directed with respect to inclusion, then by \ref{definition:set-uniform},
+    (2): If $\mathfrak{S}$ is upward-directed with respect to inclusion, then by \autoref{definition:set-uniform},
     \[
     \bracs{E(S, U)| U \in \fU, S \in \mathfrak{S}}
     \]
+
     Following the same steps in (1),
     \[
     \bracs{\bigcap_{j \in J}E(d_{j, S}, r)|J \subset I \text{ finite}, r > 0, S \in \mathfrak{S}}
     \]
+
     is a fundamental system of entourages for the $\mathfrak{S}$-uniformity.
 \end{proof}
 
@@ -108,7 +119,8 @@
     \[
     E(F, U)(f) = \bigcap_{x \in F}\pi_x^{-1}(U(f(x)))
     \]
-    which is open in the product topology. The converse is given by \ref{definition:set-uniform}.
+
+    which is open in the product topology. The converse is given by \autoref{definition:set-uniform}.
 \end{proof}
 
 \begin{proposition}