Progress over the past week.

src/topology/functions/index.tex

@@ -0,0 +1,4 @@
+\chapter{Function Spaces}
+\label{chap:function-spaces}
+
+\input{./src/topology/functions/set-systems.tex}

src/topology/functions/set-systems.tex

@@ -0,0 +1,55 @@
+\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}

src/topology/index.tex

@@ -1,5 +1,6 @@
 \part{General Topology}
-\label{part:-part-topology}
+\label{part:topology}
 
 \input{./src/topology/main/index.tex}
 \input{./src/topology/uniform/index.tex}
+\input{./src/topology/functions/index.tex}

src/topology/main/baire.tex

@@ -6,10 +6,10 @@
 \label{definition:baire}
     Let $X$ be a topological space, then the following are equivalent:
     \begin{enumerate}
-        \item For any $\seq{A_n}$ nowhere dense, $\bigcup_{n \in \nat}A_n \subsetneq X$.
-        \item For any $\seq{A_n}$ closed with empty interior, $\bigcup_{n \in \nat}A_n \subsetneq X$.
-        \item For any $\seq{A_n}$ closed with $\bigcup_{n \in \nat}A_n = X$, there exists $N \in \nat$ such that $\bigcup_{n \le N}A_n$ has non-empty interior.
-        \item For any $\seq{U_n}$ open and dense, $\bigcap_{n \in \nat}U_n$ is dense.
+        \item For any $\seq{A_n}$ nowhere dense, $\bigcup_{n \in \nat^+}A_n \subsetneq X$.
+        \item For any $\seq{A_n}$ closed with empty interior, $\bigcup_{n \in \nat^+}A_n \subsetneq X$.
+        \item For any $\seq{A_n}$ closed with $\bigcup_{n \in \nat^+}A_n = X$, there exists $N \in \nat^+$ such that $\bigcup_{n \le N}A_n$ has non-empty interior.
+        \item For any $\seq{U_n}$ open and dense, $\bigcap_{n \in \nat^+}U_n$ is dense.
     \end{enumerate}
     If the above holds, then $X$ is a \textbf{Baire space}.
 \end{definition}

src/topology/main/definition.tex

@@ -54,7 +54,7 @@
     Conversely, if $\cb \subset 2^X$ is a family such that:
     \begin{enumerate}
         \item[(TB1)] For every $x \in X$, there exists $U \in \cb$ such that $x \in U$.
-        \item[(TB2)] For every $x \in X$ and $U, V \in \cb$ such that $x \in U \cap$, there exists $W \in \cb$ such that $x \in W \subset U \cap V$.
+        \item[(TB2)] For every $x \in X$ and $U, V \in \cb$ such that $x \in U \cap V$, there exists $W \in \cb$ such that $x \in W \subset U \cap V$.
     \end{enumerate}
     then $\topo(\cb)$ is a topology on $X$, and $\cb$ is a base for $\topo(\cb)$.
 \end{definition}

src/topology/main/metric.tex

@@ -29,7 +29,7 @@
 
 \begin{lemma}
 \label{lemma:uniform-first-countable}
-    Let $X$ be a uniform space with a countable fundamental system of entourages, then there exists a countable fundamental system of symmetric entourages $\fB = \bracs{U_{1/2^n}|n \in \nat^+}$ such that $U_{1/2^{n+1}} \circ U_{1/2^{n+1}} \subset U_{1/2^n}$ for all $n \in \nat$.
+    Let $X$ be a uniform space with a countable fundamental system of entourages, then there exists a countable fundamental system of symmetric entourages $\fB = \bracs{U_{1/2^n}|n \in \nat^+}$ such that $U_{1/2^{n+1}} \circ U_{1/2^{n+1}} \subset U_{1/2^n}$ for all $n \in \nat^+$.
 \end{lemma}
 \begin{proof}
     Let $\seq{V}$ be a countable fundamental system of entourages. By \ref{lemma:symmetricfundamentalentourage}, there exists a symmetric entourage $U_{1/2} \subset V_1$. Suppose inductively that $U_{1/2^n}$ has been constructed, then by (U2) and \ref{lemma:symmetricfundamentalentourage}, there exists a symmetric entourage $U_{1/2^{n+1}} \subset U_{1/2^n} \cap V$. Thus $\fB = \bracs{U_{1/2^n}|n \in \nat^+}$ is the desired family.