On the lattice of principal generalized topologies
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On the lattice of principal generalized topologies Hassan Arianpoor1 Published online: 19 October 2018 © Akadémiai Kiadó, Budapest, Hungary 2018
Abstract In this paper we investigate the structure of the complete lattice of principal generalized topologies, employing the notion of ultratopology. On any partially ordered set we introduce a generalized topology. The existence of an anti-isomorphism between principal generalized topologies and preorder relations on a set is proposed. After determining the very basic topological properties therein, we will show that each generalized topology has a lattice complement in principal generalized topologies. Keywords Generalized topology · Principal generalized topology · Principal complement · Ultratopology Mathematics Subject Classification 54A05 · 06B30
1 Introduction and preliminaries In general topology one often studies families of subsets of a topological space; these families typically generalize the notion of open sets. The concept of generalized topology was devised by Császár [3,4], using these new open sets. Endowing the family of all generalized topologies on a set with set theoretical inclusion ⊆ one obtains a partially ordered set, in fact, a bounded lattice which is neither distributive nor complemented as discussed in [1]. Many authors studied the structure of the lattice of all topologies in a given set [8,10,16,19,20]. Gaifman [8] and Steiner [16–18] proved that the lattice is always complemented, employing the notion of ultratopology. The set of ultratopologies may naturally be devided into T1 -topologies and principal topologies. They established that the complementation problem in the lattice of principal topologies can be reduced to the sublattice of T1 -topologies. Every preorder relation is induced by a principal topology. In fact, there is a canonical isomorphism between the lattice of principal topologies and the lattice of preorder relations on a set [10,11,16]. In this paper we will show that all generalized topologies can be derived from the corresponding preorder relations and we also obtain some results by special choice of the generalized topology, the so-called principal generalized topology. Principal generalized
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Hassan Arianpoor [email protected] Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran
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topologies are then characterized by properties of generalized open sets, analogous to the usual concepts of ultratopologies introduced in [5,16]. According to Theorems (4.2) and (4.3), the lattice of principal generalized topologies on X are dually isomorphic to the lattice of preorder relations on X, and the latter is a complemented lattice. Hence, the principal generalized topologies on X form a complemented lattice, as well. Finally, the notion of complementation in the lattice of generalized topologies (Theorem 4.7) is achieved by proving that every generalized topology has a principal complement which lies in the sublattice of principal generalized topologies. Here, we recall the mo
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