The closure-complement-frontier problem in saturated polytopological spaces

Authors

  • Sara Canilang Carleton College
  • Michael P. Cohen Carleton College
  • Nicolas Graese Carleton College
  • Ian Seong Carleton College

DOI:

https://doi.org/10.53733/151

Abstract

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 $\mathcal{KF}_2$.

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Author Biographies

Sara Canilang, Carleton College

Department of Mathematics and Statistics,
Carleton College,
Northfield, MN 55057
USA

Michael P. Cohen, Carleton College

Department of Mathematics and Statistics,
Carleton College,
Northfield, MN 55057
USA

Nicolas Graese, Carleton College

Department of Mathematics and Statistics,
Carleton College,
Northfield, MN 55057
USA

Ian Seong, Carleton College

Department of Mathematics and Statistics,
Carleton College,
Northfield, MN 55057
USA

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Published

06-08-2021

How to Cite

Canilang, S., Cohen, M. P., Graese, N., & Seong, I. (2021). The closure-complement-frontier problem in saturated polytopological spaces. New Zealand Journal of Mathematics, 51, 3–27. https://doi.org/10.53733/151

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Articles