Cleanup

src/topology/main/path-connected.tex

@@ -18,10 +18,11 @@
 \begin{proof}
     Let $U, V \subset [0, 1]$ be open with $[0, 1] = U \cup V$. If $\sup U = \sup V$, then $\sup U = \sup V = 1$ and $U, V \in \cn^o(1)$, so $U \cap V \ne \emptyset$. If $\sup U < \sup V \le 1$, then $x \not\in U$ and $x \in V$. In which case, $V \in \cn^o(x)$ and $V \cap U \ne \emptyset$. Therefore $[0, 1]$ is connected.
 
-    Fix $x \in X$. For any $y \in X$, let $f_y \in C([0, 1]; X)$ be a path from $x$ to $y$, then $f_y([0, 1])$ is connected with $x, y \in f_y([0, 1])$ by \ref{proposition:connected-image}. By \ref{proposition:connected-union},
+    Fix $x \in X$. For any $y \in X$, let $f_y \in C([0, 1]; X)$ be a path from $x$ to $y$, then $f_y([0, 1])$ is connected with $x, y \in f_y([0, 1])$ by \autoref{proposition:connected-image}. By \autoref{proposition:connected-union},
     \[
     X = \bigcup_{y \in X}f_y([0, 1])
     \]
+
     is connected.
 \end{proof}
 
@@ -37,6 +38,7 @@
         g(2(t - 1/2)) &t \in [1/2, 1]
     \end{cases}
     \]
+
     is a path from $y$ to $z$.
 \end{proof}
 
@@ -45,5 +47,5 @@
     Let $X$ be a topological space and $A \subset X$ be path-connected, then there exists a unique path-connected set $C \supset A$ such that for any $C' \supset A$ path-connected, $C \supset C'$. The set $C$ is the \textbf{path-component} of $A$.
 \end{definition}
 \begin{proof}
-    Let $C$ be the union of all path-connected sets containing $A$, then $C$ is path-connected by \ref{proposition:path-connected-union} and the maximum path-connected set containing $A$ by definition.
+    Let $C$ be the union of all path-connected sets containing $A$, then $C$ is path-connected by \autoref{proposition:path-connected-union} and the maximum path-connected set containing $A$ by definition.
 \end{proof}