  Let $X$ be a space equipped with $n$ topologies $\tau_1,...,\tau_n$ 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 $\mathcal{KF}_2$.

General Topology;Combinatorics