Progress over the past week.

src/topology/functions/set-systems.tex

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+\section{Topology With Respect to Families of Sets}
+\label{section:pointwise}
+
+\begin{definition}[Set-Open Topology]
+\label{definition:set-open}
+    Let $T$ be a set, $\mathfrak{S} \subset 2^T$ be a non-empty family of sets, and $(X, \topo)$ be a topological space. For each $S \in \mathfrak{S}$ and $U \subset X$ open, let
+    \[
+    M(S, U) = \bracs{f \in X^T| f(S) \subset U}
+    \]
+    and
+    \[
+    \ce(\mathfrak{S}, \topo) = \bracs{M(S, U)| S \in \mathfrak{S}, U \in \topo}
+    \]
+    then the topology generated by $\ce$ is the \textbf{$\mathfrak{S}$-open topology} on $T^X$.
+
+    If $\cb \subset \topo$ generates $\topo$, then $\ce(\mathfrak{S}, \cb)$ generates the $\mathfrak{S}$-open topology.
+\end{definition}
+
+
+\begin{definition}[Set Uniformity]
+\label{definition:set-uniform}
+    Let $T$ be a set, $\mathfrak{S} \subset 2^T$ be a non-empty family of sets, and $(X, \fU)$ be a uniform space. For each $S \in \mathfrak{S}$ and $U \in \fU$, let
+    \[
+    E(S, U) = \bracs{(f, g) \in X^T|(f(x), g(x)) \in U \forall x \in S}
+    \]
+    and
+    \[
+    \mathfrak{E}(\mathfrak{S}, \fU) = \bracs{E(S, U)| S \in \mathfrak{S}, U \in \fU}
+    \]
+    then
+    \begin{enumerate}
+        \item $\mathfrak{E}(\mathfrak{S}, \fU)$ generates a uniformity $\fV$ on $X^T$.
+        \item The topology induced by $\fV$ is finer than the $\mathfrak{S}$-topology on $T^X$.
+        \item If $\mathfrak{S}$ is upward-directed with respect to inclusion, then $\mathfrak{E}(\mathfrak{S}, \fU)$ is forms a fundamental system of entourages for $\fV$.
+    \end{enumerate}
+    and the topology induced by $\fV$ is the \textbf{topology of uniform convergence on the sets $\mathfrak{S}$}, or the $\mathfrak{S}$-topology.
+\end{definition}
+\begin{proof}
+    (1): Since $\Delta \subset E(S, U)$ for all $S \in \mathfrak{S}$ and $U \in \fU$, $\mathfrak{E}(\mathfrak{S}, \fU)$ generates a uniformity on $X^T$.
+
+    (2): Let $U \subset X$ be open, then for each $x \in U$, there exists $V_x \in \fU$ such that $x \in V_x(x) \subset U$. In which case, $U = \bigcup_{x \in U}V_x(x)$ and $M(S, U) = \bigcup_{x \in U}M(S, V_x)(x)$.
+
+    (3): It is sufficient to verify
+    \begin{enumerate}
+        \item[(FB1)] For any $S, S'  \in \mathfrak{S}$, there exists $T \in \mathfrak{S}$ with $T \supset S, S'$. In which case, for any $U, U' \in \fU$,
+        \[
+        E(T, U \cap U') \subset E(S \cup S', U \cap U') \subset E(S, U) \cap E(S', U')
+        \]
+        \item[(FB3)] For any $U \in \fU$, there exists $V \in \fV$ with $V \circ V \subset U$. Thus for any $S \in \mathfrak{S}$,
+        \[
+        E(S, V) \circ E(S, V) \subset E(S, V \circ V) \subset E(S, U)
+        \]
+    \end{enumerate}
+    By \ref{proposition:fundamental-entourage-criterion}, $\mathfrak{E}$ is a fundamental system of entourages for the uniformity that it generates.
+\end{proof}