Polished A-A and added new lines for broken enumerates.

src/topology/main/definition.tex

@@ -9,6 +9,7 @@
         \item[(O2)] For any $U, V \in \topo$, $U \cap V \in \topo$.
         \item[(O3)] For any $\seqi{U} \subset \topo$, $\bigcup_{i \in I}U_i \in \topo$.
     \end{enumerate}
+
     The elements of $\topo$ are known as \textbf{open sets}, and the pair $(X, \topo)$ is known as a \textbf{topological space}.
 \end{definition}
 
@@ -47,6 +48,7 @@
         \item For every $x \in X$, there exists $U \in \cb$ such that $x \in U$.
         \item For every $x \in X$ and $U \subset X$ open with $x \in U$, there exists $V \in \cb$ such that $x \in V \subset U$.
     \end{enumerate}
+
     In which case,
     \[
     \topo = \topo(\cb) = \bracs{\bigcup_{i \in I}U_i \bigg | \seqi{U} \subset \cb, I \text{ index set}}
@@ -58,6 +60,7 @@
         \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 $x \in U \cap V$, 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}
 \begin{proof}
@@ -94,10 +97,10 @@
 
 \begin{definition}[Initial Topology]
 \label{definition:initial-topology}
-    Let $X$ be a set, $\bracsn{(Y_j, \topo_i)}$ be a family of topological spaces, and $\seqi{f}$ be a family of maps such that $f_i: X \to Y_i$ for each $i \in I$, then there exists a topology $\topo$ on $X$ such that:
+    Let $X$ be a set, $\bracsn{(Y_i, \topo_i)}_{i \in I}$ be a family of topological spaces, and $\seqi{f}$ be a family of maps such that $f_i: X \to Y_i$ for each $i \in I$, then there exists a topology $\topo$ on $X$ such that:
     \begin{enumerate}
         \item For each $i \in I$, $f_i \in C(X; Y_i)$.
-        \item[(U)] If $\mathcal{S}$ is a topology on $X$ satisfying $(1)$, then $\mathcal{S} \supset \topo$.
+        \item[(U)] If $\mathcal{S}$ is a topology on $X$ satisfying (1), then $\mathcal{S} \supset \topo$.
         \item The family
 
         \[
@@ -107,13 +110,49 @@
 
         is a base for $\topo$.
     \end{enumerate}
-    The topology $\topo$ is known a the \textbf{initial/weak topology} generated by the maps $\seqi{f}$.
+
+    The topology $\topo$ is the \textbf{initial/weak topology} generated by the maps $\seqi{f}$.
 \end{definition}
 \begin{proof}
-    Let $\topo$ be the topology genereated by sets of the form $\ce = \bracs{f_i^{-1}(U_i)| i \in I, U_i \in \topo_i}$. Let $\topo$ be the topology generated by $\ce$, then
+    Let $\topo$ be the topology genereated by sets of the form $\ce = \bracs{f_i^{-1}(U_i)| i \in I, U_i \in \topo_i}$, then
     \begin{enumerate}
         \item For each $i \in I$, $\topo \supset \bracs{f_i^{-1}(U)|U \in \topo_i}$, so $f_i \in C(\topo; Y_i)$.
         \item If $\mathcal{S}$ is a topology such that $f_i \in C(X, \mathcal{S}; Y_i)$, then $\bracs{f_i^{-1}(U)|U \in \topo_i} \subset \mathcal{S}$. Thus $\ce \subset \mathcal{S}$ and $\mathcal{S} \supset \topo$.
         \item By \autoref{definition:generated-topology}, $\cb$ is a base for $\topo$.
     \end{enumerate}
 \end{proof}
+
+\begin{definition}[Final Topology]
+\label{definition:final-topology}
+    Let $X$ be a set, $\bracsn{(Y_i, \topo_i)}_{i \in I}$ be a family of topological spaces, and $\seqi{f}$ be a family of maps such that $f_i: Y_i \to X$ for each $i \in I$, then there exists a topology $\topo$ on $X$ such that:
+    \begin{enumerate}
+        \item For each $i \in I$, $f_i \in C(Y_i; X)$.
+        \item[(U)] If $\mathcal{S}$ is a topology on $X$ satisfying (1), then $\mathcal{S} \subset \topo$.
+        \item For any topological space $Z$ and $F: X \to Z$, $F \in C(X; Z)$ if and only if $F \circ f_i \in C(Y_i; X)$ for all $i \in I$. 
+    \end{enumerate}
+
+    The topology $\topo$ is the \textbf{final topology} generated by the maps $\seqi{f}$.
+\end{definition}
+\begin{proof}
+    Let
+    \[
+    \topo = \bracsn{U \subset X| f_i^{-1}(U) \in \topo_i \forall i \in I}
+    \]
+
+    then since for each $i \in I$, $\topo_i$ is a topology on $Y_i$, $\topo$ is a topology on $X$. 
+
+    (1): By definition, for any $i \in I$ and $U \in \topo$, $f_i^{-1}(U) \in \topo_i$, so $f_i \in C(Y_i; X)$. 
+
+    (U): For any topology $\mathcal{S}$ satisfying (1) and $U \in \mathcal{S}$, $f_i^{-1}(U) \in \mathcal{T}_i$, so $\mathcal{S} \subset \mathcal{T}$. 
+
+    (3): Let $F: X \to Z$ such that $F \circ f_i \in C(Y_i; X)$ for all $i \in I$, then for any $U \subset Z$ open, $f_i^{-1}(F^{-1}(U)) \in \topo_i$ for all $i \in I$. Hence $F^{-1}(U) \in \topo$ and $F \in C(X; Z)$. 
+\end{proof}
+
+\begin{definition}[Generated Topology]
+\label{definition:ideal-generated-topology}
+    Let $X$ be a topological space and $\sigma \subset 2^X$ be an ideal, then $X$ is \textbf{$\sigma$-generated} if the topology of $X$ is the final topology generated by $\bracs{\iota_S: S \to X|S \in \sigma}$. 
+
+    If $\kappa \subset 2^X$ is the collection of precompact sets of $X$, and $X$ is generated by $\kappa$, then $X$ is \textbf{compactly generated}. 
+\end{definition}
+
+