Boundary-Border Extensions of the Kuratowski Monoid

TL;DR

This paper explores the structure and size of extended Kuratowski monoids in topology, introducing the concept of Kuratowski disconnected spaces and analyzing how boundary operators affect these monoids.

Contribution

It characterizes the boundary-border extensions of the Kuratowski monoid, introduces Kuratowski disconnected spaces, and investigates the collapse behaviors of the Gaida-Eremenko monoid.

Findings

When $| ext{K}|<14$, the GE monoid is determined by $ ext{K}$.

If $| ext{K}|=14$ and certain conditions hold, $| ext{KF}|=28$; otherwise, $| ext{KF}|=34$.

The study identifies 70 possible collapse behaviors of the GE monoid.

Abstract

The Kuratowski monoid $\mathbf{K}$ is generated under operator composition by closure and complement in a nonempty topological space. It satisfies $2\leq|\mathbf{K}|\leq14$. The Gaida-Eremenko (or GE) monoid $\mathbf{KF}$ extends $\mathbf{K}$ by adding the boundary operator. It satisfies $4\leq|\mathbf{KF}|\leq34$. We show that when $|\mathbf{K}|<14$ the GE monoid is determined by $\mathbf{K}$. When $|\mathbf{K}|=14$ if the interior of the boundary of every subset is clopen, then $|\mathbf{KF}|=28$. This defines a new type of topological space we call $Kuratowski\ disconnected$. Otherwise $|\mathbf{KF}|=34$. When applied to an arbitrary subset the GE monoid collapses in one of $70$ possible ways. We investigate how these collapses and $\mathbf{KF}$ interdepend, settling two questions raised by Gardner and Jackson. Computer experimentation played a key role in our research.