Added the Mackey-Arens theorem.

src/topology/main/compactify.tex

@@ -104,7 +104,7 @@
         \item $\infty \in U$ and $U^c \subset X$ is compact. 
     \end{enumerate}
 
-    Let $\seqi{U} \subset \mathcal{T}^*$ be an open cover of $X$, then there exists $i \in I$ such that $\infty \in U$. In which case, $U_i$ must satisfy (b), so there exists $J \subset I$ finite such that $\bigcup_{j \in J}U_j \supset U_i^c$, and $\bracsn{U_j|j \in J \cup \bracs{i}}$ is a finite subcover. Now, let $x \in X$, then since $X$ is locally compact, there exists a precompact neighbourhood $U \in \cn_X^o(x)$. In which case, $\ol{U}^c \in  \cn_{X^*}(\infty)$ with $U \cap \ol{U}^c = \emptyset$. Therefore $X^*$ is a compact Hausdorff space. 
+    Let $\seqi{U} \subset \mathcal{T}^*$ be an open cover of $X$, then there exists $i \in I$ such that $\infty \in U$. In which case, $U_i$ must satisfy (b), so there exists $J \subset I$ finite such that $\bigcup_{j \in J}U_j \supset U_i^c$, and $\bracsn{U_j|j \in J \cup \bracs{i}}$ is a finite subcover. Now, let $x \in X$, then since $X$ is locally compact, there exists a relatively compact neighbourhood $U \in \cn_X^o(x)$. In which case, $\ol{U}^c \in  \cn_{X^*}(\infty)$ with $U \cap \ol{U}^c = \emptyset$. Therefore $X^*$ is a compact Hausdorff space. 
     
     Let $\iota: X \to X^*$ be the inclusion map. For each $U \in \mathcal{T}^*$ satisfying (b), $\iota^{-1}(U) = U \cap X$. Since $U^c \subset X$ is compact, $U \cap X$ is open by \autoref{proposition:compact-closed}, so $\iota \in C(X; X^*)$. By (a), $\iota$ is an embedding.