The lattices of families of regular sets in topological spaces

Author:

Przemska Emilia1

Affiliation:

1. Institute of Mathematics , Pomeranian University , 76-200 , Slupsk , Poland

Abstract

Abstract 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, ∨, ∧}, where b, i, ∨ and ∧ denote the closure operator, the interior operator, the binary operators corresponding to union and intersection, respectively, is called the Kuratowski {b, i, ∨, ∧}-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.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Reference23 articles.

1. Al-Hassani, O.—Mahesar, Q. A.—Sacerdoti Coen, C.—Sorge, V.: A Term Rewriting System for Kuratowski’s Closure-Complement Problem, 23rd International Conference on Rewriting Techniques and Applications. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2012.

2. Andrijević, D.: On b-open sets, Mat. Vesnik (1996), 59–64.

3. Aull, C. E.: Classification of topological spaces, Bulletin De ľAcademie Polonaise des Sciences-Serie des Sciences Mathematiques Astronomiques et Physiques, 15(11) (1967), 773–778.

4. Banakh, T.—Chervak, O.—Martynyuk, T.—Pylypovych, M.—Ravsky, A.—Simkiv, M.: Kuratowski monoids of n-topological spaces, Topol. Algebra Appl. 6(1) (2015), 1–25.

5. Birkhoff, G.: Lattice Theory. Amer. Math. Soc. Coll. Publ, 1967.

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