Added bornological spaces.

src/fa/tvs/bounded.tex

@@ -3,13 +3,13 @@
 
 \begin{definition}[Bounded]
 \label{definition:bounded}
-    Let $E$ be a TVS over $K \in \RC$ and $B \subset E$, then the following are equivalent:
+    Let $(E, \topo)$ be a TVS over $K \in \RC$ and $B \subset E$, then the following are equivalent:
     \begin{enumerate}
         \item For every $U \in \cn(0)$, there exists $\lambda \in K$ such that $\lambda U \supset B$. 
         \item For every $\seq{x_n} \subset B$ and $\seq{\lambda_n} \subset K$ such that $\lambda_n \to 0$, $\lambda_n x_n \to 0$ as $n \to \infty$.
     \end{enumerate}
 
-    If the above holds, then $B$ is \textbf{bounded}. The collection $B(E)$ is the set of all bounded sets of $E$.
+    If the above holds, then $B$ is \textbf{bounded}. The collection $B(E) = B(E, \topo)$ is the set of all bounded sets of $E$.
 \end{definition}
 \begin{proof}
     (1) $\Rightarrow$ (2): Let $U \in \cn_E(0)$ be circled, then there exists $k \in \natp$ such that $kU \supset B$. Since $\lambda_n \to 0$ as $n \to \infty$, there exists $N \in \natp$ such that $|\lambda_n| \le 1/k$ for all $n \ge N$. In which case, $\lambda_n x_n \in \lambda_n B \subset U$ for all $n \ge N$. 

src/fa/tvs/complete-metric.tex

@@ -111,7 +111,7 @@
     \end{enumerate}
 \end{theorem}
 \begin{proof}
-    Suppose that $T(E)$ is not meagre. Let $r_0 > 0$ and $r > 0$ such that $B_E(0, r) + B_E(0, r) \subset B_E(0, r_0)$, then since $B_E(0, r)$ is absorbing,
+    Suppose that $T(E)$ is not meagre. Let $r_0 > 0$ and $r > 0$ such that $B_E(0, r) + B_E(0, r) \subset B_E(0, r_0)$, then since $B_E(0, r)$ is radial,
     \[
     E = \bigcup_{n \in \natp}nB_E(0, r) \quad \overline{T(E)} = \bigcup_{n \in \natp}\overline{nT(B_E(0, r))}
     \]

src/fa/tvs/definition.tex

@@ -137,7 +137,7 @@
 \label{proposition:tvs-good-neighbourhood-base}
     Let $E$ be a topological vector space over $K \in \RC$, then
     \begin{enumerate}
-        \item $E$ admits a fundamental system of neighbourhoods at $0$ consisting of circled and absorbing sets.
+        \item $E$ admits a fundamental system of neighbourhoods at $0$ consisting of circled and radial sets.
         \item The fundamental system of neighbourhoods in $(1)$ can be taken to be open or closed.
     \end{enumerate}
 \end{proposition}

src/fa/tvs/projective.tex

@@ -20,6 +20,7 @@
 
         is a fundamental system of neighbourhoods for $E$ at $0$.
     \end{enumerate}
+    
     The uniformity $\fU$ and its topology are the \textbf{projective uniformity/topology} induced by $\seqi{T}$.
 \end{definition}
 \begin{proof}