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- W4288280890 abstract "Let $X$ be a space equipped with $n$ topologies $tau_1,...,tau_n$ which are pairwise comparable and saturated, and for each $1leq ileq n$ 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 ${k_i,f_i:1leq ileq n}cup{c}$ (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 $Asubseteq X$ 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 $mathcal{KF}_2$." @default.
- W4288280890 created "2022-07-28" @default.
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- W4288280890 date "2019-07-18" @default.
- W4288280890 modified "2023-10-01" @default.
- W4288280890 title "The closure-complement-frontier problem in saturated polytopological spaces" @default.
- W4288280890 doi "https://doi.org/10.48550/arxiv.1907.08203" @default.
- W4288280890 hasPublicationYear "2019" @default.
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