Finished basic function space topologies.

src/topology/functions/set-systems.tex

@@ -53,3 +53,21 @@
     \end{enumerate}
     By \ref{proposition:fundamental-entourage-criterion}, $\mathfrak{E}$ is a fundamental system of entourages for the uniformity that it generates.
 \end{proof}
+
+\begin{definition}[Pointwise Topology]
+\label{definition:pointwise}
+    Let $T$ be a set and $X$ be a topological space, then the following topologies on $X^T$ coincide:
+    \begin{enumerate}
+        \item The product topology on $X^T$.
+        \item The $\mathfrak{S}$-open topology, where $\mathfrak{S} = \bracs{\bracs{x}| x \in X}$.
+        \item (If $X$ is a uniform space) The $\mathfrak{F}$-uniform topology, where $\fF = \bracs{F| F \subset X \text{ finite}}$.
+    \end{enumerate}
+    This topology is the \textbf{topology of pointwise convergence} on $X^T$.
+\end{definition}
+\begin{proof}
+    (2) $=$ (3): Let $F \subset X$ finite and $U$ be an entourage, $f \in X^T$, then
+    \[
+    E(F, U)(f) = \bigcap_{x \in F}\pi_x^{-1}U(f(x))
+    \]
+    which is open in the product topology. The converse is given by \ref{definition:set-uniform}.
+\end{proof}