Added the one-point compactification.

src/topology/main/compactify.tex

@@ -71,5 +71,59 @@
     
 \end{proof}
 
+\begin{lemma}
+\label{lemma:lch-compactification-open}
+    Let $X$ be a LCH space and $(Y, \varphi)$ be a compactification of $X$, then $\varphi(X) \subset Y$ is open. 
+\end{lemma}
+\begin{proof}
+    For each $x \in X$, let $U \in \cn_X(x)$ be a compact neighbourhood. Since $Y$ is a compact Hausdorff space, $\varphi(U)$ is closed by \autoref{proposition:compact-closed}. As $\varphi \in C(X; Y)$ is an embedding, there exists $V \in \cn_Y(\varphi(x))$ such that $\varphi(U) = \varphi(X) \cap V$. Given that $\varphi(X)$ is dense in $Y$, $\varphi(U) = \ol{\varphi(X) \cap V} \supset V$. Therefore $\varphi(U) \in \cn_{Y}(\varphi(x))$, and $\varphi(X)$ is open in $Y$. 
+\end{proof}
+
+
+
+\begin{definition}[One-Point Compactification]
+\label{definition:alexandroff-compactification}
+    Let $(X, \mathcal{T})$ be a LCH space, then there exists a pair $(X^*, \iota)$ such that:
+    \begin{enumerate}
+        \item $(X^*, \iota)$ is a compactification of $X$. 
+        \item[(U)] For any pair $(Y, \varphi)$ satisfying (1), there exists a unique $\varphi^* \in C(Y; X^*)$ such that the following diagram commutes:
+        \[
+        \xymatrix{
+        Y \ar@{->}[rd]^{\varphi^*} &  \\
+        X \ar@{->}[r]_{\iota} \ar@{->}[u]^{\varphi} & X^*
+        }
+        \]
+    \end{enumerate}
+    
+    The pair $(X^*, \iota)$ is the \textbf{one-point compactification} of $X$.
+\end{definition}
+\begin{proof}
+    Let $\infty$ be a point not in $X$, $X^* = X \sqcup \bracs{\infty}$, and $\mathcal{T}^* \subset 2^{X^*}$ such that for each $U \in \mathcal{T}^*$, one of the following holds:
+    \begin{enumerate}[label=(\alph*)]
+        \item $U \in \mathcal{T}$.
+        \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 $\iota: X \to X^*$ 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. 
+
+    Finally, let
+    \[
+    \varphi^*: Y \to X^* \quad x \mapsto \begin{cases}
+        \varphi^{-1}(x) &x \in \varphi(X) \\
+        \infty &x \not\in \varphi(X)
+    \end{cases}
+    \]
+
+    Let $U \subset X^*$ with $\infty \not\in U$, then $(\varphi^*)^{-1}(U) = \varphi(U)$ is open in $\varphi(X)$ because $\varphi$ is an embedding, and open in $Y$ by \autoref{lemma:lch-compactification-open}. On the other hand, for each $V \in \cn_{X^*}^o(\infty)$, 
+    \[
+    (\varphi^*)^{-1}(V) = V \cup (Y \setminus \varphi(X))
+    \]
+
+    Since $\varphi \in C(X; Y)$ is an embedding, $V$ is relatively open in $\varphi(X)$, so $V \cup (Y \setminus \varphi(X))$ is open in $Y$. 
+\end{proof}
+
+