Added uniform structures for completely regular spaces. Added calculus lemma.

src/topology/main/compactify.tex

@@ -0,0 +1,75 @@
+\section{Compactifications}
+\label{section:compactifications}
+
+\begin{definition}[Compactification]
+\label{definition:compactification}
+    Let $X$ be a topological space, then a \textbf{compactification} of $X$ is a pair $(Y, f)$ where
+    \begin{enumerate}
+        \item $Y$ is a compact Hausdorff space. 
+        \item $f \in C(X; Y)$ is an embedding.
+        \item $f(X)$ is dense in $Y$. 
+    \end{enumerate}
+\end{definition}
+
+\begin{definition}[Stone-Čech Compactification]
+\label{definition:stone-cech}
+    Let $X$ be a completely regular space, then there exists a pair $(\beta X, e)$ such that:
+    \begin{enumerate}
+        \item $(\beta X, e)$ is a compactification of $X$. 
+        \item[(U1)] For any $f \in C(X; [0, 1])$, there exists a unique $\beta f \in C(\beta X; [0, 1])$ such that the following diagram commutes:
+        \[
+        \xymatrix{
+        \beta X \ar@{->}[r]^{\beta f} & [0, 1] \\
+        X \ar@{->}[u]^{e} \ar@{->}[ru]_{f} & 
+        }
+        \]
+    \end{enumerate}
+
+    Moreover, if $(\beta X, e)$ is \textit{any} pair that satisfies (1) and (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:
+        \[
+        \xymatrix{
+        \beta X \ar@{->}[r]^{\beta \varphi} & Y \\
+        X \ar@{->}[u]^{e} \ar@{->}[ru]_{\varphi} & 
+        }
+        \]
+    \end{enumerate}
+
+    The pair $(\beta X, e)$ is the \textbf{Stone-Čech compactification} of $X$. 
+\end{definition}
+\begin{proof}
+    Let $e: X \to [0, 1]^{C(X; [0, 1])}$ be the embedding of $X$ into $[0, 1]^{C(X; [0, 1])}$ associated with $C(X; [0, 1])$ in \autoref{definition:embedding-in-cube}, and $\beta X = \ol{e(X)}$. 
+
+    (1): By \autoref{theorem:tychonoff} and \autoref{proposition:product-hausdorff}, $[0, 1]^{C(X; [0, 1])}$ is a compact Hausdorff space. By definition, $e(X)$ is dense in $\beta X$. 
+
+    (U1): For each $f \in C(X; [0, 1])$, $\pi_f \in C([0, 1]^{C(X; [0, 1])}; [0, 1])$ is an extension of $f$ to $e(x)$. 
+
+    (U2): Let $(Y, \varphi)$ be a compactification of $X$. For each $f \in C(Y; [0, 1])$, by (U1), there exists a unique $\beta(f \circ \varphi) \in C(X; [0, 1])$ such that the following diagram commutes: 
+    \[
+    \xymatrix{
+    X \ar@{->}[d]_{\varphi} \ar@{->}[r]^{e} & \beta X \ar@{->}[d]^{\beta (f \circ \varphi)} \\
+    Y \ar@{->}[r]_{f} & [0, 1]
+    }
+    \]
+
+    Let $e': Y \to [0, 1]^{C(Y; [0, 1])}$ be the embedding of $Y$ into $[0, 1]^{C(Y; [0, 1])}$ associated with $C(Y; [0, 1])$, then by (U) of the \hyperref[product topology]{definition:product-topology}, there exists $\beta(e' \circ \varphi) \in C(\beta X; [0, 1])$ such that the following diagram commutes:
+    \[
+    \xymatrix{
+    X \ar@{->}[d]_{\varphi} \ar@{->}[r]^{e} & \beta X \ar@{->}[d]^{\beta (e' \circ \varphi)} \\
+    Y \ar@{->}[r]_{e'} & [0, 1]^{C(Y; [0, 1])}
+    }
+    \]
+
+    Since $Y$ is a compact Hausdorff space, $e'(Y)$ is closed by \autoref{proposition:compact-extensions} and \autoref{proposition:compact-closed}. As $e'$ is an embedding, identify $Y$ as a subspace of $[0, 1]^{C(Y; [0, 1])}$. Given that $e(X)$ is dense in $\beta X$, the the image of $\beta (e' \circ \varphi)$ lies in $Y$ by \autoref{proposition:closure-of-image}. Therefore under the identification, the following diagram commutes:
+    \[
+    \xymatrix{
+    X \ar@{->}[d]_{\varphi} \ar@{->}[r]^{e} & \beta X \ar@{->}[ld]^{\beta (e' \circ \varphi)} \\
+    Y & 
+    }
+    \] 
+    
+\end{proof}
+
+
+

src/topology/main/cube.tex

@@ -0,0 +1,63 @@
+\section{Embeddings in Cubes}
+\label{section:embeddings-in-cubes}
+
+\begin{definition}[Completely Regular]
+\label{definition:completely-regular}
+    Let $X$ be a topological space, then $X$ is \textbf{completely regular} if for any $E \subset X$ closed and $x \in X \setminus E$, there exists $f \in C(X; [0, 1])$ such that $f(x) = 1$ and $f|_E = 0$. 
+\end{definition}
+
+\begin{definition}[Separation of Points and Closed Sets]
+\label{definition:separate-points-closed-sets}
+    Let $X$ be a topological space and $\cf \subset C(X; [0, 1])$, then $\cf$ \textbf{separates points and closed sets} if for any $E \subset X$ closed and $x \in X \setminus E$, there exists $f \in \cf$ such that $f(x) \not\in \ol{f(E)}$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:completely-regular-separate}
+    Let $X$ be a $T_1$ space, then the following are equivalent:
+    \begin{enumerate}
+        \item $X$ is completely regular. 
+        \item There exists $\cf \subset C(X; [0, 1])$ that separates points and closed sets.
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (2) $\Rightarrow$ (1): Let $E \subset X$ closed and $x \in X \setminus E$, then there exists $f \in \cf$ such that $x \not\in \ol{f(E)}$. By \hyperref[Urysohn's lemma]{lemma:urysohn}, there exists $\phi \in C([0, 1]; [0, 1])$ such that $\phi(f(x)) = 1$ and $\phi(f(E)) = 0$. 
+\end{proof}
+
+\begin{definition}[Embedding in Cube]
+\label{definition:embedding-in-cube}
+    Let $X$ be a topological space, $\cf \subset C(X; [0, 1])$, and
+    \[
+    e: X \to [0, 1]^\cf \quad \pi_f(e(x)) = f(x)
+    \]
+
+    then:
+    \begin{enumerate}
+        \item $e \in C(X; [0, 1]^\cf)$. 
+        \item If $\cf$ separates points, then $e$ is injective.
+        \item If $X$ is $T_1$ and $\cf$ separates points and closed sets, then $e$ is an embedding. 
+    \end{enumerate}
+
+    The mapping $e$ is the \textbf{mapping of $X$ into the cube $[0, 1]^\cf$ associated with $\cf$}. 
+\end{definition}
+\begin{proof}[Proof, {{\cite[Proposition 4.53]{Folland}}}. ]
+    (1): By (U) of the \hyperref[product topology]{definition:product-topology}. 
+
+    (3): Since $X$ is $T_1$, $e$ is injective by (2). Let $x \in X$ and $U \in \cn_X^o(x)$, then there exists $f \in \cf$ such that $f(x) \not\in \ol{f(U^c)}$. In which case, there exists $V \in \cn_{[0, 1]}^o(f(x))$ such that $V \cap f(U^c) = \emptyset$. Thus for any $y \in X$ with $\pi_f(e(y)) \in V$, $f(y) \not\in f(U^c)$, so $y \in U$. 
+\end{proof}
+
+\begin{proposition}
+\label{proposition:completely-regular-uniformisable}
+    Let $X$ be a $T_1$ space, then the following are equivalent:
+    \begin{enumerate}
+        \item $X$ is completely regular.
+        \item There exists a uniformity $\fU$ on $X$ that induces the topology on $X$. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1) $\Rightarrow$ (2): By \autoref{definition:uniform-separated}. 
+
+    (2) $\Rightarrow$ (1): By \autoref{definition:embedding-in-cube}, $X$ embeds into $[0, 1]^{C(X; [0, 1])}$, which is a uniform space. The subspace uniformity on $X$ then induces the topology on $X$. 
+\end{proof}
+
+
+

src/topology/main/index.tex

@@ -24,3 +24,5 @@
 \input{./c0.tex}
 \input{./semicontinuity.tex}
 \input{./baire.tex}
+\input{./cube.tex}
+\input{./compactify.tex}