Me when I forget to commit.

src/dg/derivative/sets.tex

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+\section{Differentiation With Respect to Set Systems}
+\label{section:differentiation-set-system}
+
+\begin{definition}[Small]
+\label{definition:differentiation-small}
+    Let $E, F$ be TVSs over $K \in \RC$, $\sigma \subset B(E)$ be an upward-directed family of sets that contains all finite sets, $r: E \to F$, and $n \in \natz$, then the following are equivalent:
+    \begin{enumerate}
+        \item For each $A \in \sigma$, $r(th)/t^n \to 0$ uniformly on $A$.
+        \item If $r_t(x) = r(tx)/t^n$, then $r_t \to 0$ as $t \to 0$ with respect to the $\sigma$-uniform topology on $F^E$.
+        \item For each $A \in \sigma$, $\seq{a_k} \subset A$, and $\seq{t_k} \subset K \setminus \bracs{0}$ with $t_k \to 0$ as $n \to \infty$, $r(t_ka_k)/t_k^n \to 0$ as $n \to \infty$.
+    \end{enumerate}
+    If the above holds, then $r$ is \textbf{$\sigma$-small of order $n$}.
+\end{definition}
+
+\begin{remark}
+    In \ref{definition:differentiation-small}, the system $\sigma$ can be chosedn based on the bornology of $E$, and the definition of small-ness depends exclusively on $\sigma$. As such, there is an apparent disconnect between differentiation and the topology of the domain.
+\end{remark}