A Kuratowski closure-complement variant whose solution is independent of ZF

TL;DR

This paper investigates a new variant of the Kuratowski closure-complement problem involving the $d$ operator in Polish spaces, revealing that the number of obtainable sets varies depending on the underlying set theory system.

Contribution

It introduces a novel Kuratowski variant with the $d$ operator and determines the exact number of distinct sets in different set-theoretic frameworks.

Findings

In ZFC, the maximum number of distinct sets is 22.

In ZF+DC+PB, the maximum number reduces to 18.

The $d$ operator's role affects the closure-complement set hierarchy.

Abstract

We pose the following new variant of the Kuratowski closure-complement problem: How many distinct sets may be obtained by starting with a set $A$ of a Polish space $X$, and applying only closure, complementation, and the $d$ operator, as often as desired, in any order? The set operator $d$ was studied by Kuratowski in his foundational text \textit{Topology: Volume I}; it assigns to $A$ the collection $dA$ of all points of second category for $A$. We show that in ZFC set theory, the answer to this variant problem is $22$. In a distinct system equiconsistent with ZFC, namely ZF+DC+PB, the answer is only $18$.