1966/04/01 by A. K. Steiner · 1 citation
Mathematics · Computer Science · #Rings, Modules, and Algebras #Advanced Topology and Set Theory #Advanced Algebra and Logic #Comparison of topologies #Network topology #Mathematics #Lattice (music) #Countable set #Complementation #Discrete mathematics #Topology (electrical circuits) #Combinatorics #Topological space #Extension topology #General topology #Computer science #Physics
paper · pdf · doi:10.2307/1994555
openalex publication_date 1966/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/04/21
The family of all topologies on a set is a complete, bounded lattice.The purpose of this paper is to study the structure of the lattice of topologies, employing the notion of ultraspace introduced by Frhlich [7] and to show that this lattice is complemented.The set of ultraspaces may naturally be divided into two classes, each of which generates a sublattice.One of these sublattices is the lattice of -topologies.The other, which is studied in 2, is the lattice of principal topologies.Principal topologies are defined in terms of ultraspaces and are then characterized by properties of open sets.Some topological properties of ultraspaces are investigated in 4 and maximal regular (maximal Ty, maximal normal, etc.) topologies are characterized in terms of ultraspaces.The problem of complementation in the lattice of topologies has been outstanding for some time.However, several partial solutions have been provided.Hartmanis [11] first showed the lattice was complemented if the ground set was finite and asked whether this was true in the infinite case.Gaifman [8] gave a positive answer for denumerable sets and Berri [3], using this fact, was able to provide complements for certain special topologies such as a topological group with a dense, nonopen countable subgroup.It is shown in 5 that the lattice of principal topologies is complemented.Gaifman [9] established that the complementation problem can be reduced to verifying that each Tropology has a complement.In 6, it is proved that if every T,-topology has a lattice complement which is a principal topology, then every topology does.In 7 it is shown that the lattice of topologies on an arbitrary set is complemented by proving that every topology in the sublattice of Ty -topologies has a lattice complement which lies in the sublattice of principal topologies.Preliminary definitions and remarks.Throughout this paper, E will denote an arbitrary set, and t, with or without subscripts, will denote a topology on E,