Added facts about vector measures.

src/topology/uniform/cauchy.tex

@@ -24,6 +24,21 @@
     Let $(X, \fU)$ be a uniform space and $\fF \subset 2^X$ be a filter on $X$, then $\fF$ is \textbf{Cauchy} if for every $V \in \fU$, there exists $E \in \fF$ such that $E$ is $V$-small.
 \end{definition}
 
+\begin{definition}[Cauchy Net]
+\label{definition:cauchy-net}
+    Let $(X, \fU)$ be a uniform space and $\net{x} \subset X$ be a net, then $\net{x}$ is \textbf{Cauchy} if for any $V \in \fU$, there exists $\alpha_0 \in A$ such that $(x_\alpha, x_\beta) \in V$ for all $\alpha, \beta \ge \alpha_0$.
+\end{definition}
+
+\begin{lemma}
+\label{lemma:cauchy-subnet}
+    Let $(X, \fU)$ be a uniform space and $\net{x} \subset X$ be a Cauchy net. If there exists a subnet $\angles{x_\beta}_B \subset \net{x}$ and $x \in X$ such that $x_\beta \to x$, then $x_\alpha \to x$.
+\end{lemma}
+\begin{proof}
+    Let $U \in \fU$ and $V \in \fU$ such that $V \circ V \subset U$, then there exists $\alpha_0 \in A$ such that $(x_\alpha, x_{\alpha'}) \in V$ for all $\alpha, \alpha' \ge \alpha_0$, and $\beta \in B$ with $\beta \ge \alpha_0$ such that $(x_\beta, x) \in V$. Thus, $(x_\alpha, x) \in V \circ V \subset U$ for all $\alpha \ge \alpha_0$, so $x_\alpha \to x$.
+\end{proof}
+
+
+
 \begin{proposition}[Cauchy Criterion]
 \label{proposition:cauchycriterion}
     Let $(X, \fU)$ be a uniform space and $\fF \subset 2^X$ be a convergent filter, then $\fF$ is Cauchy.