The lattices of families of regular sets in topological spaces

Year, pages: 2020, 477 - 488

The question as to the number of sets obtainable from a given subset of a topological space using the operators derived by composing members of the set $\{b, i, \vee, \wedge\}$, where $b$, $i$, $\vee$ and $\wedge$ denote the closure operator, the interior operator, the binary operators corresponding to union and intersection, respectively, is called the Kuratowski $\{b, i, \vee, \wedge\}$-problem. This problem has been solved independently by Sherman [21] and, Gardner and Jackson [13], where the resulting $34$ plus identity operators were depicted in the Hasse diagram. In this paper we investigate the sets of fixed points of these operators. We show that there are at most $23$ such families of subsets. Twelve of them are the topology, the family of all closed subsets plus, well known generalizations of open sets, plus the families of their complements. Each of the other $11$ families forms a complete complemented lattice under the operations of join, meet and negation defined according to a uniform procedure. Two of them are the well known Boolean algebras formed by the regular open sets and regular closed sets, any of the others in general need not be a Boolean algebras.

Przemska, E. 2020. The lattices of families of regular sets in topological spaces. In Mathematica Slovaca, vol. 70, no.2, pp. 477-488. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0365

Przemska, E. (2020). The lattices of families of regular sets in topological spaces. Mathematica Slovaca, 70(2), 477-488. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0365

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava