Added some example facts.

src/topology/main/compact.tex

@@ -70,6 +70,15 @@
     Let $x \in E^c$. For each $y \in E$, there exists $U_y \in \cn(y)$ and $V_y \in \cn(x)$ such that $U_y \cap V_y = \emptyset$. By compactness, there exists $E_0 \subset E$ finite such that $\bigcup_{y \in E_0}U_y \supset E$. In which case, $E \cap \bigcap_{y \in E_0}V_y = \emptyset$ and $\bigcap_{y \in E_0}V_y \in \cn(x)$. Thus $E^c$ is open.
 \end{proof}
 
+\begin{proposition}
+\label{proposition:compact-hausdorff-homeomorphism}
+    Let $X$ be a compact topological space, $Y$ be a Hausdorff space, and $f \in C(X; Y)$ be a bijection, then $f$ is a homeomorphism. 
+\end{proposition}
+\begin{proof}
+    For each $K \subset X$ closed, $K$ and $f(K)$ are compact by \autoref{proposition:compact-extensions}. By \autoref{proposition:compact-closed}, $f(K)$ is closed. Thus $f^{-1}$ maps closed sets to closed sets, and hence open sets to open sets.
+\end{proof}
+
+
 \begin{proposition}
 \label{proposition:compact-hausdorff-normal}
     Let $X$ be a Hausdorff space, $A, B \subset X$ be compact with $A \cap B = \emptyset$, then there exists $U \in \cn(A)$ and $V \in \cn(B)$ such that $U \cap V = \emptyset$.

src/topology/main/compactify.tex

@@ -25,7 +25,7 @@
         \]
     \end{enumerate}
 
-    Moreover, if $(\beta X, e)$ is \textit{any} pair that satisfies (1) and (U1), then
+    Moreover, if $(\beta X, e)$ is \textit{any} pair that satisfies (U1), then
     \begin{enumerate}
         \item[(U2)] For any pair $(Y, \varphi)$ satisfying (1), there exists a unique $\beta \varphi \in C(\beta X; Y)$ such that the following diagram commutes:
         \[