Cleanup

src/topology/main/definition.tex

@@ -51,6 +51,8 @@
     \[
     \topo = \topo(\cb) = \bracs{\bigcup_{i \in I}U_i \bigg | \seqi{U} \subset \cb, I \text{ index set}}
     \]
+
+
     Conversely, if $\cb \subset 2^X$ is a family such that:
     \begin{enumerate}
         \item[(TB1)] For every $x \in X$, there exists $U \in \cb$ such that $x \in U$.
@@ -65,6 +67,8 @@
     \[
     \paren{\bigcup_{i \in I}U_i} \cap \paren{\bigcup_{j \in J}V_j} = \bigcup_{i \in I}\bigcup_{j \in J}U_i \cap V_j
     \]
+
+
     Thus to show that $\topo(\cb)$ satisfies (O2), it is sufficient to show that $U \cap V \in \topo(\cb)$ for all $U, V \in \cb$. To this end, for every $x \in U \cap V$, there exists $W_x \in \cb$ such that $x \in W_x \subset U \cap V$. Therefore $U \cap V = \bigcup_{x \in U \cap V}W_x \in \topo(\cb)$, and $\topo(\cb)$ satisfies (O2).
 
     By definition, $\topo(\cb)$ satisfies (O3).
@@ -76,14 +80,16 @@
     \[
     \topo(\ce) = \bracs{\bigcup_{i \in I}U_i \bigg | U_i \in \cb(\ce)}
     \]
+
     where
     \[
     \cb(\ce) = \bracs{\bigcap_{j = 1}^n U_j \bigg | \seqf{U_j} \subset \ce, n \in \nat^+}
     \]
+
     is a base for $\topo(\ce)$. The topology $\topo(\ce)$ is known as the topology \textbf{generated by} $\ce$.
 \end{definition}
 \begin{proof}
-    Since $\cb(\ce) \supset \ce$, $\cb(\ce)$ satisfies (TB1). In addition, $\cb(\ce)$ is closed under finite intersections, so it satisfies (TB2). By \ref{definition:base}, $\cb(\ce)$ is a base for $\topo(\ce)$.
+    Since $\cb(\ce) \supset \ce$, $\cb(\ce)$ satisfies (TB1). In addition, $\cb(\ce)$ is closed under finite intersections, so it satisfies (TB2). By \autoref{definition:base}, $\cb(\ce)$ is a base for $\topo(\ce)$.
 \end{proof}
 
 \begin{definition}[Initial Topology]
@@ -98,6 +104,7 @@
         \mathcal{B} = \bracs{\bigcap_{j \in J}f_j^{-1}(U_j) \bigg | J \subset I \text{ finite}, U_j \in \topo_j}
         \]
 
+
         is a base for $\topo$.
     \end{enumerate}
     The topology $\topo$ is known a the \textbf{initial/weak topology} generated by the maps $\seqi{f}$.
@@ -107,6 +114,6 @@
     \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 \ref{definition:generated-topology}, $\cb$ is a base for $\topo$.
+        \item By \autoref{definition:generated-topology}, $\cb$ is a base for $\topo$.
     \end{enumerate}
 \end{proof}