What is the "co-derived set" in topology?

Definition 2.9 in this paper introduces a co-derived set operator $f$ (in the paper the operator is denoted by $\mathbf{Cod}$). The function $f:\mathcal P(X)\longrightarrow\mathcal P(X)$ is characterized by the following properties:

• $f(X) = X$,
• $x\in f(A)$ iff $x\in f(A\cup\{x\})$,
• $A\cap f(A)\subseteq f(A\cap f(A))$, and
• $f(A\cap B) = f(A)\cap f(B)$

for each $A,B\in\mathcal P(X)$.

My question is now: what is the co-derived set operator? More precisely, there is also derived set operator $g$ (Definition 2.6 in the paper; in the paper the derived set operator is denoted by $\mathbf{Der}$, but it is also common to use a prime to denote the derived set). The derived set operator is a function $g:\mathcal P(X)\longrightarrow\mathcal P(X)$ that is characterized by

• $g(\emptyset) = \emptyset$,
• $x\in g(A)$ iff $x\in g(A\setminus\{x\})$,
• $g(A\cup g(A)) \subseteq A\cup g(A)$, and
• $g(A\cup B) = g(A)\cup g(B)$

for each $A,B\in\mathcal P(X)$.

For this operator I already know that it corresponds to the set of all limit points of a set $A$. That is, $g(A)$ is the union of all points $x\in X$ such that for every $U\in\mathcal T$ with $x\in U$ it holds that $A\cap (U\setminus\{x\}) \neq \emptyset$, i. e. , $$g(A) = \big\{x\in X : \text{$A\cap (U\setminus\{x\})\neq\emptyset$ for all $U\in\mathcal T$ with $x\in U$}\big\}.$$ I am looking for a similar description of $f(A)$. Does $f(A)$ describe the set of all interior points of $A$?

Solution 1:

The axioms for $f$ are constructed from those of $g$ (which is the derived set operator) by replacing $\cup$ by $\cap$ and vice versa, interchanging $\emptyset$ and $X$ and reversing the inclusions where appropriate. So its axioms are dual to those of the derived set and that's why it's named the co-derived set operator I think.

They're dual in the same way as closure and interior operators are, if you look at their axioms.

I think that if $g$ is a derived set operator, $f(A)=g(A^\complement)^\complement$ will be a co-derived set operator and vice versa. The paper mentions that too, I saw.

Just like a set is closed iff it contains its derived set, we have the dual fact that a set is open iff its a subset of its coderived set. I don't think an alternative description is very enlightening. It's a completely artificial "Spielerei".