Added Urysohn Metrisation theorem and compactness theorems.

src/topology/metric/metric.tex

@@ -3,7 +3,7 @@
 
 \begin{definition}[Metric Space]
 \label{definition:metric}
-    Let $X$ be a set and $d: X \times X$, then $d$ is a \textbf{metric} if:
+    Let $X$ be a set and $d: X \times X \to [0, \infty]$, then $d$ is a \textbf{metric} if:
     \begin{enumerate}
         \item[(PM1)] For any $x \in X$, $d(x, x) = 0$.
         \item[(M)] For any $x, y \in X$ with $x \ne y$, $d(x, y) > 0$.
@@ -41,10 +41,23 @@
     (3) $\Rightarrow$ (1): Let $\seq{x_n} \subset X$ be a countable dense subset. Let $x \in X$ and $k \in \natp$, then there exists $x_n \in \natp$ such that $d(x, x_n) < 1/(2k)$. In which case, $x \in B(x_n, 1/(2k)) \subset B(x_n, 1/k)$. Therefore $\bracs{B(x_n, 1/k)|n, k \in \natp}$ forms a countable basis for $X$. 
 \end{proof}
 
+\begin{proposition}
+\label{proposition:countable-metric}
+    Let $\seq{(X_n, d_n)}$ be metrisable spaces, then $\prod_{n \in \natp}X_n$ is also metrisable. 
+\end{proposition}
+\begin{proof}
+    For each $n \in \natp$, let
+    \[
+    d_n': \braks{\prod_{n \in \natp}X_n}^2 \to [0, \infty] \quad (x, y) \mapsto d_n(\pi_n(x), \pi_n(y))
+    \]
+
+    then $d_n'$ is a pseudometric on $X$, and $\bracsn{d_n'}_1^\infty$ induces the product uniformity on $\prod_{n \in \natp}X_n$. By \autoref{theorem:uniform-metrisable}, $\prod_{n \in \natp}X_n$ is also metrisable. 
+\end{proof}
+
 
 \begin{theorem}[Banach's Fixed Point Theorem]
 \label{theorem:banach-fixed-point}
-    Let $(X, d)$ be a metric space and $f: X \to X$. If there exists $C \in (0, 1)$ such that
+    Let $(X, d)$ be a complete metric space and $f: X \to X$. If there exists $C \in (0, 1)$ such that
     \[
     d(f(x), f(y)) \le Cd(x, y) \quad \forall x, y \in X
     \]