TY - JOUR
AU - Canilang, Sara
AU - Cohen, Michael P.
AU - Graese, Nicolas
AU - Seong, Ian
PY - 2021/08/06
Y2 - 2024/08/07
TI - The closure-complement-frontier problem in saturated polytopological spaces
JF - New Zealand Journal of Mathematics
JA - NZ J Math
VL - 51
IS -
SE - Articles
DO - 10.53733/151
UR - https://nzjmath.org/index.php/NZJMATH/article/view/151
SP - 3--27
AB - <p>Let $X$ be a space equipped with $n$ topologies $\tau_1,\ldots,\tau_n$ which are pairwise comparable and saturated, and for each $1\leq i\leq n$ 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 $\mathcal{KF}_n$ generated by $\{k_i,f_i:1\leq i\leq n\}\cup\{c\}$ (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 $\mathcal{KF}_2$.</p>
ER -