Added Urysohn Metrisation theorem and compactness theorems.

src/topology/main/definition.tex

@@ -77,6 +77,18 @@
     By definition, $\topo(\cb)$ satisfies (O3).
 \end{proof}
 
+\begin{definition}[First Countable]
+\label{definition:first-countable}
+    Let $X$ be a topological space, then $X$ is \textbf{first countable} if for every $x \in X$, there exists a countable fundamental system of neighbourhoods at $x$. 
+\end{definition}
+
+\begin{definition}[Second Countable]
+\label{definition:second-countable}
+    Let $X$ be a topological space, then $X$ is \textbf{second countable} if it admits a countable base. 
+\end{definition}
+
+
+
 \begin{definition}[Generated Topology]
 \label{definition:generated-topology}
     Let $X$ be a set and $\ce \subset 2^X$ such that $\bigcup_{U \in \ce}U = X$, then the smallest topology on $X$ containing $\ce$ is given by