Recently I stumbled upon the concept of residue of a set (alternately known as the frontier of a set) in general topology, which is the closure of a set A without A itself, or in other words all of the boundary points of A not contained in A: $$\operatorname{Res}(A)= \overline A \setminus A$$ I'm not quite sure who first came up with this idea, so I'm not able to provide any useful references. This is very similar to the boundary of A, which for context is usually defined as the closure minus the interior: $$ \partial(A)= \overline A \setminus A^{\circ}$$ I'm trying to find applications for this concept, so I've been looking for some properties of this residue operation, such as how unions and intersections of sets behave under this operation, how many different sets I can obtain using only the union and residue operations, whether I can use this notion as a primitive for defining topological spaces or weaker notions of it, etc. So far I have the following: \begin{align} \operatorname{Res}(A) \subseteq & \partial A \\ \operatorname{Res}(C)=&\varnothing \Leftrightarrow \text{C is closed} \\ \operatorname{Res}(U)=&\partial(U) \Leftrightarrow \text{U is open} \\ \complement_X{(\operatorname{Res}(A))}&=A \cup \operatorname{Ext}(A) \\ \operatorname{Res}\left(\operatorname{Res}(A)\right)&\cap \operatorname{Res}(A)= \varnothing \quad * \end{align} *(hence each successive iteration of $\operatorname{Res}$ will be disjoint from the previous one.)
Just to be absolutely clear, here I define the exterior of A as: $$\operatorname{Ext}(A)=\complement_X{\left(\overline A\right)=\left(\complement_X{(A)}\right)^\circ}$$
This is pretty tedious work, and I realize to many this is pretty trivial and not all that interesting, but I just want to see how much I can get out of this residue operator and whether it has any interesting connections to other topological notions.
More of a response to the comments than a direct answer to the question, although the comments suggested this could count as an application:
You can get arbitrarily many sets by iterating the residue operator.
If $X_1$ is the unit circle minus a point then $\operatorname{Res} X_1$ is a point.
If $X_2=D\setminus X_1$, where D is the unit disc, then $\operatorname{Res} X_2 = X_1$. So iterating the residue operator gives you four sets ($X_2$,$X_1$, point, empty set).
You can build more sets $X_n$ in an analogous manner, getting arbitrarily large numbers of sets you can reach by iterated residues alone.
These sorts of things come up with lexicographic orders, so you might want to see if there's some useful connection.