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

src/topology/main/baire.tex

@@ -9,7 +9,8 @@
         \item For any $\seq{A_n} \subset 2^X$ nowhere dense, $\bigcup_{n \in \nat^+}A_n \subsetneq X$.
         \item For any $\seq{A_n} \subset 2^X$ closed with empty interior, $\bigcup_{n \in \nat^+}A_n$ has empty interior.
         \item For any $\seq{U_n} \subset 2^X$ open and dense, $\bigcap_{n \in \nat^+}U_n$ is dense.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $X$ is a \textbf{Baire space}.
 \end{definition}
 \begin{proof}
@@ -28,7 +29,8 @@
     \begin{enumerate}
         \item[(a)] For all $n > 1$, $\ol V_n \subset U_n \cap V_{n - 1} \subset U$.
         \item[(b)] $\bigcap_{j \in \natp} \ol V_j$ is non-empty.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $X$ is a Baire space.
 \end{lemma}
 \begin{proof}

src/topology/main/compact.tex

@@ -9,7 +9,8 @@
         \item For every family $\seqi{E}$ of closed sets with $\bigcap_{j \in J}E_j \ne \emptyset$ for all $J \subset I$ finite, $\bigcap_{i \in I}E_i \ne \emptyset$.
         \item Every filter in $X$ has a cluster point.
         \item Every ultrafilter in $X$ converges.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $X$ is \textbf{compact}.
 \end{definition}
 \begin{proof}
@@ -89,7 +90,8 @@
     \begin{enumerate}
         \item For any $x \in X$ and $U \in \cn_{X \times Y}^o(\bracs{x} \times Y)$, there exists $V \in \cn_X(x)$ such that $V \times Y \subset U$. 
         \item For any $A \subset X$ and $U \in \cn_{X \times Y}^o(A \times Y)$, there exists $V \in \cn_X(A)$ such that $V \times Y \subset U$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{lemma}
 \begin{proof}

src/topology/main/connected.tex

@@ -8,7 +8,8 @@
         \item For any $\emptyset \ne U, V \subset X$ open with $U \cup V = X$, $U \cap V \ne \emptyset$.
         \item There exists no surjective $f \in C(X; \bracs{0, 1})$.
         \item For any $U \subset X$ open and closed, either $U = \emptyset$ or $U = X$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $X$ is \textbf{connected}.
 \end{definition}
 \begin{proof}

src/topology/main/continuity.tex

@@ -8,7 +8,8 @@
     \begin{enumerate}
         \item For each $V \in \cn(f(x))$, $f^{-1}(V) \in \cn(x)$.
         \item For each filter base $\fB \subset 2^X$ converging to $x$, $f(\fB)$ converges to $f(x)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $f$ is \textbf{continuous at} $x \in X$.
 
     The following are also equivalent:
@@ -16,7 +17,8 @@
         \item For each $U \subset Y$ open, $f^{-1}(U)$ is open in $X$.
         \item $f$ is continuous at every $x \in X$.
         \item For each convergent filter base $\fB \subset 2^X$, $f(\fB)$ is convergent.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $f$ is \textbf{continuous}.
 
     The collection $C(X; Y)$ is the space of all continuous functions from $X$ to $Y$.
@@ -39,7 +41,8 @@
     \begin{enumerate}
         \item[(a)] $\bigcup_{i \in I}U_i = X$.
         \item[(b)] For each $i, j \in I$, either $U_i \cap U_j = \emptyset$, or $f_i|_{U_i \cap U_j} = f_j|_{U_i \cap U_j}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a unique $f \in C(X; Y)$ such that $f|_{U_i} = f_i$ for all $i \in I$.
 \end{lemma}
 \begin{proof}

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}
+
+

src/topology/main/filters.tex

@@ -25,7 +25,8 @@
     \begin{enumerate}
         \item[(FB1)] For any $E, F \in \fB$, there exists $G \in \fB$ such that $G \subset E \cap F$.
         \item[(FB2)] $\emptyset \not\in \fB$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Conversely, if $\fB \subset 2^X$ is a non-empty collection that satisfies (FB1) and (FB2), then $\fB$ is a base for the filter
     \[
     \fF = \bracs{F \subset X| \exists E \in \fB: E \subset F}
@@ -52,7 +53,8 @@
     \begin{enumerate}
         \item $f(\fB) = \bracs{f(E)| E \in \fB}$ is also a filter base.
         \item If $\fB$ is an ultrafilter base, then $f(\fB)$ is also an ultrafilter base. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}
@@ -117,7 +119,8 @@ The smallest filter $\fF(\fB_0) \subset 2^X$ containing $\fB_0$ is the filter \t
         \item $\fF$ is maximal with respect to inclusion.
         \item For any $E \subset X$, either $E \in \fF$ or $E^c \in \fF$.
         \item For any $\seqf{F_j} \subset X$ such that $\bigcup_{j = 1}^n F_j \in \fF$, there exists $1 \le j \le n$ such that $F_j \in \fF$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $\fF$ is an \textbf{ultrafilter}.
 \end{definition}
 \begin{proof}
@@ -157,7 +160,8 @@ The smallest filter $\fF(\fB_0) \subset 2^X$ containing $\fB_0$ is the filter \t
         \item $\fF \supset \cn(x)$.
         \item For each ultrafilter $\fU \supset \fF$, $\fU \supset \cn(x)$.
         \item There exists a fundamental system of neighbourhoods $\cb(x) \subset \cn(x)$ such that for every $E \in \cb$, there exists $F \in \fB$ with $F \subset E$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $x$ is a \textbf{limit point} of $\fB$, and $\fB$ \textbf{converges} to $x$.
 
     If $A \subset X$ and $\fB \subset 2^A$, then $\fB$ \textbf{converges} to $x$ if $\fF(\fB) \supset \bracsn{U \cap A| U \in \cn(x)}$.
@@ -178,7 +182,8 @@ The smallest filter $\fF(\fB_0) \subset 2^X$ containing $\fB_0$ is the filter \t
         \item $x \in \bigcap_{E \in \fF}\overline{E}$.
         \item There exists a fundamental system of neighbourhoods $\cb(x) \subset \cn(x)$ such that for every $E \in \cb$ and $f \in \fB$, $E \cap F \ne \emptyset$.
         \item There exists a filter $\fU \supset \fB$ that converges to $x$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $x$ is a \textbf{cluster/accumulation point} of $\fB$. In particular, if $\fF$ is an ultra filter, then (6) implies that the limit points and cluster points of $\fF$ coincide.
 \end{definition}
 \begin{proof}

src/topology/main/hausdorff.tex

@@ -12,7 +12,8 @@
         \item Every filter in $X$ converges to at most one point.
         \item For any index set $I$, the diagonal $\Delta$ is closed in $X^I$.
         \item The diagonal $\Delta$ is closed in $X \times X$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $X$ is a \textbf{T2/Hausdorff} space.
 \end{definition}
 \begin{proof}

src/topology/main/lch.tex

@@ -8,7 +8,8 @@
         \item For any $x \in X$, there exists $K \in \cn(x)$ compact.
         \item For any $x \in X$, $\cn(x)$ admits a fundamental system of neighbourhoods consisting of compact sets.
         \item For any $x \in X$, $\cn(x)$ admits a fundamental system of neighbourhoods consisting of precompact sets.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $X$ is a \textbf{locally compact Hausdorff (LCH)} space.
 \end{definition}
 \begin{proof}
@@ -76,6 +77,35 @@
     then by the \hyperref[gluing lemma for continuous functions]{lemma:gluing-continuous}, $\ol F \in C_c(X; \real)$ with $\ol F|_K = F|_K = f$ and $\supp{F} \subset \supp{\eta} \subset V \subset U$.
 \end{proof}
 
+\begin{proposition}
+\label{proposition:lch-compactly-generated}
+    Let $X$ be a LCH space, then:
+    \begin{enumerate}
+        \item $X$ is compactly generated. 
+        \item For any uniform space $Y$, $C(X; Y) \subset Y^X$ is closed with respect to the compact-open topology. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1): Let $U \subset X$ such that $U \cap K$ is open in $K$ for all $K \subset X$ compact. For any $x \in U$, there exists a compact neighbourhood $K \in \cn(x)$. In which case, $U \supset U \cap K \in \cn(x)$, so $U \in \cn(x)$ for all $x \in U$. By \autoref{lemma:openneighbourhood}, $U$ is open. 
+
+    (2): By \autoref{proposition:compact-uniform-open}, the compact-open topology coincides with the compact-uniform topology on $C(X; Y)$. Since $X$ is compactly generated, $C(X; Y) \subset Y^X$ is closed with respect to the compact-open topology by \autoref{corollary:uniform-limit-continuous-generated}. 
+\end{proof}
+
+
+\begin{proposition}
+\label{proposition:lch-product}
+    Let $\seqi{X}$ be a family of LCH spaces. If $X_i$ is compact for all but finitely many $i \in I$, then $X = \prod_{i \in I}X_i$ is a LCH space. 
+\end{proposition}
+\begin{proof}
+    By \autoref{proposition:product-hausdorff}, $\prod_{i \in I}X_i$ is Hausdorff. Let $x \in \prod_{i \in I}X_i$ and $i \in I$. If $X_i$ is not compact, let $U_i \in \cn_{X_i}(\pi_i(x))$ be compact. Otherwise, let $U_i = X_i$. Let $U = \prod_{i \in I}U_i$, then since $U_i \ne X_i$ for only finitely many $i \in I$, $U \in \cn_X(x)$. By \hyperref[Tychonoff's Theorem]{theorem:tychonoff}, $U$ is compact. Therefore $X$ is locally compact. 
+\end{proof}
+
+
+\subsection{Paracompactness and LCH Spaces}
+\label{subsection:lch-paracompact}
+
+
+
 \begin{proposition}[{{\cite[Proposition 4.39]{Folland}}}]
 \label{proposition:lch-sigma-compact}
     Let $X$ be a LCH space, then the following are equivalent:
@@ -92,7 +122,8 @@
         \item[(a)] For each $0 \le k \le n$, $U_k$ is a precompact open set.
         \item[(b)] For each $0 \le k < n$, $\overline{U_k} \subset U_{k+1}$.
         \item[(c)] For each $1 \le k \le n$, $U_k \supset \bigcup_{j = 1}^k K_j$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By \autoref{lemma:lch-compact-neighbour}, there exists $U_{n+1} \in \cn^o(\overline{U_n} \cup K_{n+1})$ precompact. In which case, by (c),
     \[
     U_{n+1} \supset \ol{U_n} \cup K_{n+1} \supset \bigcup_{j = 1}^n K_j \cup K_{n+1} = \bigcup_{j = 1}^{n+1}K_j
@@ -154,7 +185,8 @@
     \begin{enumerate}
         \item[(a)] $\ol{E} \subset \bigcup_{x \in X_E}N_x$.
         \item[(b)] For every $x \in X_E$, $N_x \cap E \ne \emptyset$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Let $X_{\ce} = \bigcup_{E \in \ce}X_\ce$, and for each $E \in \ce$, let
     \[
     G_E = \bigcup_{\substack{x \in X_\ce \\ \ol{N_x} \subset E}}N_x
@@ -238,12 +270,4 @@
     Let $x \in X$, then there exists $n \in \natp$ such that $x \in U_n \setminus U_{n-1}$. In which case, if $n > 1$, then $x \in U_{n} \setminus \ol{U_{n - 2}} = V_{n-1}$. If $n = 1$, then $x \in U_{2} = V_1$. Thus $\seq{V_n}$ is an open cover of $X$. In addition, for any $m, n \in \natp$ with $m \le n$, $V_m \cap V_n \ne \emptyset$ implies that $n - m < 2$, so $\seq{V_n}$ is locally finite. By (2) of \autoref{proposition:lch-paracompact}, $X$ is paracompact.
 \end{proof}
 
-\begin{proposition}
-\label{proposition:lch-product}
-    Let $\seqi{X}$ be a family of LCH spaces. If $X_i$ is compact for all but finitely many $i \in I$, then $X = \prod_{i \in I}X_i$ is a LCH space. 
-\end{proposition}
-\begin{proof}
-    By \autoref{proposition:product-hausdorff}, $\prod_{i \in I}X_i$ is Hausdorff. Let $x \in \prod_{i \in I}X_i$ and $i \in I$. If $X_i$ is not compact, let $U_i \in \cn_{X_i}(\pi_i(x))$ be compact. Otherwise, let $U_i = X_i$. Let $U = \prod_{i \in I}U_i$, then since $U_i \ne X_i$ for only finitely many $i \in I$, $U \in \cn_X(x)$. By \hyperref[Tychonoff's Theorem]{theorem:tychonoff}, $U$ is compact. Therefore $X$ is locally compact. 
-\end{proof}
-
 

src/topology/main/neighbourhoods.tex

@@ -36,12 +36,14 @@
         \item[(F2)] For any $A, B \in \cn_\topo(x)$, $A \cap B \in \cn_\topo(x)$.
         \item[(V1)] For every $A \in \cn_\topo(x)$, $x \in A$.
         \item[(V2)] For every $V \in \cn_\topo(x)$, there exists $W \in \cn_\topo(x)$ such that $V \in \cn_\topo(y)$ for all $y \in W$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Conversely, if $\cn: X \to 2^X$ is a mapping such that
     \begin{enumerate}
         \item $\cn(x) \ne \emptyset$ for all $x \in X$.
         \item $\cn(x)$ satisfies (F1), (F2), (V1), and (V2).
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a unique topology $\topo \subset 2^X$ such that $\cn = \cn_\topo$.
 \end{proposition}
 

src/topology/main/normal.tex

@@ -23,7 +23,8 @@
         \begin{enumerate}
             \item[(a)] $U_1 = B^c$.
             \item[(b)] For any $p, q \in \rational \cap [0, 1]$ with $p < q$, $\overline{U_p} \subset U_q$.
-        \end{enumerate}
+        \end\{enumerate\}
+
         \item There exists $f \in C(X; [0, 1])$ with $f|_A = 0$ and $f|_B = 1$.
     \end{enumerate}
 \end{lemma}

src/topology/main/product.tex

@@ -40,7 +40,7 @@
 
 \begin{proposition}
 \label{proposition:productfilterconvergence}
-Let $\bracsn{(X_i, \topo_i)}_{i \in I}$ be a family of topological spaces and $\B$ be a filter base on $\prod_{i \in I}X_i$, then $\fB$ converges to $x \in \prod_{i \in I}X_i$ if and only if $\pi_i(\fB)$ converges to $\pi_i(x)$ for all $i \in I$.
+Let $\bracsn{(X_i, \topo_i)}_{i \in I}$ be a family of topological spaces and $\fB$ be a filter base on $\prod_{i \in I}X_i$, then $\fB$ converges to $x \in \prod_{i \in I}X_i$ if and only if $\pi_i(\fB)$ converges to $\pi_i(x)$ for all $i \in I$.
 \end{proposition}
 \begin{proof}
     $(\Rightarrow)$: Let $i \in I$ and $U \in \cn(\pi_i(x))$, then $\pi_i^{-1}(U) \in \cn(x)$. Since $\fB$ converges to $x$, there exists $B \in \fB$ with $B \subset \pi_i^{-1}(U)$. In which case, $\pi_i(B) \subset U$ and $\pi_i(\fB)$ converges to $\pi_i(x)$.

src/topology/main/quotient.tex

@@ -12,7 +12,8 @@
     \begin{enumerate}
         \item For any $U \subset Y$, $U$ is open if and only if $\pi^{-1}(U)$ is open.
         \item $\pi \in C(X; Y)$, and for any $U \subset X$ saturated and open, $\pi(U)$ is open.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $\pi$ is a \textbf{quotient map}.
 \end{definition}
 \begin{proof}
@@ -37,7 +38,8 @@
         \]
 
         \item $\pi$ is a quotient map.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The space $(\td X, \pi)$ is the \textbf{quotient} of $X$ by $\sim$.
 \end{definition}
 \begin{proof}

src/topology/main/regular.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item For each $x \in X$ and $A \subset X$ closed with $x \not\in A$, there exists $U \in \cn(x)$ and $V \in \cn(A)$ such that $U \cap V = \emptyset$.
         \item For each $x \in X$, the closed neighbourhoods of $x$ forms a fundamental system of neighbourhoods at $x$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If $X$ is a T1 space such that the above holds, then $X$ is \textbf{regular}.
 \end{definition}
 \begin{proof}[Proof {{\cite[Proposition 1.4.11]{Bourbaki}}}. ]