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 $\mathcal{KF}_2$.
Highlights
Following the tradition for Kuratowski-type problems, we exhibit an explicit initial set in R, equipped with the usual and Sorgenfrey topologies, which yields 2p(2) = 120 distinct sets under the action of the monoid kyfz ∈ (KF) 2. In his 1922 thesis [8], Kuratowski posed and solved the following problem: given a topological space (X, τ ), what is the largest number of distinct subsets that can be obtained by starting from an initial set A ⊆ X, and applying the topological closure and complement operators, in any order, as often as desired? The answer is 14
If we equip a space X with not one but two distinct topologies τ1 and τ2, how many distinct subsets may be obtained by starting with an initial set, and applying each of the two associated closure operators k1, k2, and the set complement operator c, in any order, as often as desired? The authors construct an example of a bitopological space (X, τ1, τ2) where it is possible to obtain infinitely many subsets from a certain initial set
It was shown independently by Gardner and Jackson [6] and by Sherman [11] that in any topological space (X, τ ), the greatest number of sets one may obtain from an initial set A ⊆ X by applying the set operators {k, i, ∪, ∩} is 35
Summary
In his 1922 thesis [8], Kuratowski posed and solved the following problem: given a topological space (X, τ ), what is the largest number of distinct subsets that can be obtained by starting from an initial set A ⊆ X, and applying the topological closure and complement operators, in any order, as often as desired? The answer is 14. 1, number of points needed for is four, while the minimal number of points needed to contain a 34-set is 8 Another interesting question that remains open is to solve the closure-complementfrontier problem for polytopological spaces which are not necessarily saturated. It would be interesting to study some of the variants described in Section 4 of [6] in the larger context of polytopological spaces It was shown independently by Gardner and Jackson [6] and by Sherman [11] that in any topological space (X, τ ), the greatest number of sets one may obtain from an initial set A ⊆ X by applying the set operators {k, i, ∪, ∩} is 35. What is the largest number of sets one may obtain from an initial set A ⊆ X by applying the set operators kj, ij (1 ≤ j ≤ n), ∪, and ∩ in any order, as often as desired?
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