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| Main Authors: | , , , |
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| Format: | Preprint |
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2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1907.08203 |
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| _version_ | 1866911162164051968 |
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| author | Canilang, Sara Cohen, Michael P. Graese, Nicolas Seong, Ian |
| author_facet | Canilang, Sara Cohen, Michael P. Graese, Nicolas Seong, Ian |
| contents | Let $X$ be a space equipped with $n$ topologies $τ_1,...,τ_n$ 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 $(X,τ_1,...,τ_n)$ 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 $\mathcal{KF}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_08203 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The closure-complement-frontier problem in saturated polytopological spaces Canilang, Sara Cohen, Michael P. Graese, Nicolas Seong, Ian General Topology Combinatorics 54A10, 54E55, 06F05 Let $X$ be a space equipped with $n$ topologies $τ_1,...,τ_n$ 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 $(X,τ_1,...,τ_n)$ 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 $\mathcal{KF}_2$. |
| title | The closure-complement-frontier problem in saturated polytopological spaces |
| topic | General Topology Combinatorics 54A10, 54E55, 06F05 |
| url | https://arxiv.org/abs/1907.08203 |