The closure-complement-frontier problem in saturated polytopological spaces

Abstract

Let $X$ be a space equipped with $n$ topologies ${\tau }_1,...,{\tau }_n$ which are pairwise comparable and saturated, and for each $1\le i\le n$ let $k_i$ and $f_i$ be the associated topological closure and frontier operators, respectively. Inspired by the closure-complement theorem of Kuratowski, we prove that the monoid of set operators ${\mathcal{K}\mathcal{F}}_n$ generated by $\{k_i,f_i:1\le i\le n\}\cup \{c\}$ (where $c$ denotes the set complement operator) has cardinality no more than $2p(n)$ where $p(n)=\frac{5}{24}{n}^4+\frac{37}{12}{n}^3+\frac{79}{24}{n}^2+\frac{101}{12}n+2$. The bound is sharp in the following sense: for each $n$ there exists a saturated polytopological space $(X,{\tau }_1,...,{\tau }_n)$ and a subset $A\subseteq X$ such that repeated application of the operators $k_i,f_i,c$ to $A$ will yield exactly $2p(n)$ distinct sets. In particular, following the tradition for Kuratowski-type problems, we exhibit an explicit initial set in $\mathbb{R}$, equipped with the usual and Sorgenfrey topologies, which yields $2p(2)=120$ distinct sets under the action of the monoid ${\mathcal{K}\mathcal{F}}_2$.

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References in corpus (1)

• Kuratowski's Theorem for Two Closure Operators

Cited by in corpus (2)

• A Kuratowski closure-complement variant whose solution is independent of ZF
• Boundary-Border Extensions of the Kuratowski Monoid