Adjusted citation formats. Moved citation off of named theorems if possible.

src/topology/main/lch.tex

@@ -35,11 +35,11 @@
     so $V = \bigcup_{j = 1}^n V_{x_j} \in \cn^o(K)$ is precompact.
 \end{proof}
 
-\begin{lemma}[Urysohn's Lemma (LCH), {{\cite[Lemma 4.32]{Folland}}}]
+\begin{lemma}[Urysohn's Lemma (LCH)]
 \label{lemma:lch-urysohn}
     Let $X$ be a LCH space, $K \subset X$ be compact, and $U \in \cn(K)$, then there exists $f \in C_c(X; [0, 1])$ such that $\supp{f} \subset U$.
 \end{lemma}
-\begin{proof}
+\begin{proof}[Proof {{\cite[Lemma 4.32]{Folland}}}. ]
     By \autoref{lemma:lch-compact-neighbour}, there exists $V, W \in \cn^o(K)$ precompact such that
     \[
     K \subset V \subset \ol{V} \subset W \subset \ol{W} \subset U

src/topology/main/regular.tex

@@ -1,7 +1,7 @@
 \section{Regular Spaces}
 \label{section:regularspaces}
 
-\begin{definition}[Regular Space, {{\cite[Proposition 1.4.11]{Bourbaki}}}]
+\begin{definition}[Regular Space]
 \label{definition:regular}
     Let $X$ be a topological space, then the following are equivalent:
     \begin{enumerate}
@@ -10,7 +10,7 @@
     \end{enumerate}
     If $X$ is a T1 space such that the above holds, then $X$ is \textbf{regular}.
 \end{definition}
-\begin{proof}
+\begin{proof}[Proof {{\cite[Proposition 1.4.11]{Bourbaki}}}. ]
     $(1) \Rightarrow (2)$: Let $U \in \cn^o(x)$, then $U^c$ is closed with $x \not\in U^c$ by (V1). Thus there exists $V \in \cn^o(U^c)$ such that $x \not\in V$, so $V^c \in \cn(x)$ is closed with $V^c \subset U$.
 
     $(2) \Rightarrow (1)$: Since $X$ is Hausdorff, $X$ is T1 as well. Let $x \in X$ and $A \subset X$ closed such that $x \not\in A$, then $A^c \in \cn(x)$. So there exists $K \in \cn(x)$ closed such that $x \in K \subset A^c$. Thus $K \in \cn(x)$ and $K^c \in \cn(A)$ are the desired neighbourhoods.

src/topology/uniform/complete.tex

@@ -48,7 +48,7 @@
 \end{proof}
 
 
-\begin{theorem}[Extension of Uniformly Continuous Functions, {{\cite[Theorem 1.3.2]{Bourbaki}}}]
+\begin{theorem}[Extension of Cauchy Continuous Functions]
 \label{theorem:uniform-continuous-extension}
     Let $(X, \fU)$ be a uniform space, $(Y, \fV)$ be a complete Hausdorff uniform space, $A \subset Y$ be a dense subset, and $f \in C(A; Y)$ be Cauchy continuous, then:
     \begin{enumerate}
@@ -56,7 +56,7 @@
         \item If $f \in UC(A; Y)$, then $F \in UC(X; Y)$.
     \end{enumerate}
 \end{theorem}
-\begin{proof}
+\begin{proof}[Proof {{\cite[Theorem 1.3.2]{Bourbaki}}}. ]
     (1): By \autoref{proposition:imagecauchy}, $f(\bracs{U \cap A| U \in \cn(x)})$ is a Cauchy filter base for all $x \in X$. Thus the existence and uniqueness of the extension follows from \autoref{proposition:uniformextension}.
 
     (2): Let $V \in \fV$. Using \autoref{proposition:goodentourages}, assume without loss of generality that $V$ is symmetric and closed. Since $f \in UC(A; Y)$, there exists $U \in \fU$ such that $(f(x), f(y)) \in V$ whenever $(x, y) \in U$. Using \autoref{proposition:subspace-entourage}, assume without loss of generality that $U = \overline{U} = \ol{U \cap (A \times A)}$. In which case, since $F$ is continuous,

src/topology/uniform/completion.tex

@@ -2,7 +2,7 @@
 \label{section:completion}
 
 
-\begin{definition}[Hausdorff Completion, {{\cite[Theorem 2.3.3]{Bourbaki}}}]
+\begin{definition}[Hausdorff Completion]
 \label{definition:hausdorff-completion}
     Let $(X, \fU)$ be a uniform space, then there exists a $(\wh X, \iota)$ such that:
     \begin{enumerate}
@@ -35,7 +35,7 @@
 
     The pair $(\wh X, \iota)$ is the \textbf{Hausdorff completion} of $X$.
 \end{definition}
-\begin{proof}
+\begin{proof}[Proof {{\cite[Theorem 2.3.3]{Bourbaki}}}. ]
     (1, Uniform), (4): Let $\wh X$ be the set of all minimal Cauchy filters on $X$. For each $V \in \fU$, let
     \[
     \wh V = \bracsn{(\fF, \mathfrak{G}) \in \wh X| \exists U \in \fF \cap \mathfrak{G}: U \times U \subset V}
@@ -89,7 +89,7 @@
     Since the mapping $(\iota \times \iota)^{-1}$ is a bijection between two bases of uniformities of $X$ and $\wh X$, it is sufficient to show that $\iota$ is injective. To this end, observe that for any $x, y \in X$, $\cn(x) = \cn(y)$ if and only if $x = y$ by (4) of \autoref{definition:hausdorff}.
 \end{proof}
 
-\begin{definition}[Associated Hausdorff Uniform Space, {{\cite[Proposition 2.8.16]{Bourbaki}}}]
+\begin{definition}[Associated Hausdorff Uniform Space]
 \label{definition:associated-hausdorff}
     Let $(X, \fU)$ be a uniform space, then there exists $(X', i)$ such that:
     \begin{enumerate}
@@ -107,7 +107,7 @@
     \end{enumerate}
     known as the \textbf{Hausdorff uniform space associated with} $(X, \fU)$.
 \end{definition}
-\begin{proof}
+\begin{proof}[Proof {{\cite[Proposition 2.8.16]{Bourbaki}}}. ]
     Let $(\wh X, \iota)$ be the Hausdorff completion of $X$, $X' = \iota(X)$, and $i = \iota$, then $(X', i)$ satisfies (1) and (2).
 
     (U): Let $(\wh Y, \iota)$ be the Hausdorff completion of $Y$. Using \autoref{lemma:completion-of-hausdorff}, identify $Y$ as a subspace of $\wh Y$. By (U) of the Hausdorff completion, there exists a unique $\ol F \in UC(\wh X; \wh Y)$ such that the following diagram commutes:

src/topology/uniform/definition.tex

@@ -113,7 +113,7 @@
     so $\fb_S$ is a fundamental system of entourages.
 \end{proof}
 
-\begin{definition}[Topology of a Uniform Space, {{\cite[Proposition 2.1.2]{Bourbaki}}}]
+\begin{definition}[Topology of a Uniform Space]
 \label{definition:uniformtopology}
     Let $(X, \fU)$ be a uniform space and
     \[
@@ -122,7 +122,7 @@
 
     then there exists a unique topology $\topo \subset 2^X$ such that $\cn_\topo = \cn$, known as the \textbf{topology induced by the uniform structure $\fU$}.
 \end{definition}
-\begin{proof}
+\begin{proof}[Proof {{\cite[Proposition 2.1.2]{Bourbaki}}}. ]
 Using \autoref{proposition:neighbourhoodcharacteristic}, it is sufficient to show that $\cn(x)$ is non-empty for all $x \in X$, and that it satisfies (F1), (F2), (V1), and (V2). Firstly, since $\fU \ne \emptyset$, $\cn(x) \ne \emptyset$ for all $x \in X$.
 
 (F1): Let $U \in \fU$ and $V \supset U(x)$, then