Complementation in the lattice of $T\sb{1}$-topologies

Abstract

Introduction. The purpose of this paper is to study the complementation problem in the lattice of Ti-topologies. In ?1 it is shown that a large class of T1-topologies do have complements. However, in general, the lattice of Ti-topologies is not a complemented lattice; a counterexample will be presented in ?2. Let I be the family of all topologies definable on an arbitrary set E. For riE2 and r2(E, ri<T2 if every set in ri is in 12. Then ri is said to be coarser than T2 and T2 finer than ri. Under this order, 2 is a complete lattice. The greatest element of 2 is the discrete topology, 1, and the least element is the trivial topology, 0. A topology with the property that the only finer topology is the discrete topology, is called an ultraspace on E. The collection (E of subsets of E consisting of P(E { x }) UJ, where xCE, i is a filter on E, and P(E{x}) is the power set of E{x}, is a topology, denoted (5(x, ). Fr6hlich [2] proved that there is a one-to-one correspondence between ultraspaces on E and topologies of the form S(x, W); where xCE and W is an ultrafilter on E, different from the principal ultrafilter at x, t(x). An ultraspace 25(x, t) is a Ti-topology if and only if 2t is a nonprincipal ultrafilter. In this case, 2t contains no finite sets and S(x, 2t) is called a nonprincipal ultraspace. A topology on E is a Ti-topology if and only if it is the infimum of nonprincipal ultraspaces. Since any topology finer than a Ti-topology is a Ti-topology, the family A of Ti-topologies is a complete sublattice of the lattice of all topologies. The lattice A has a greatest element, 1, and a least element, the cofinite topology C, in which the empty set and complements of finite sets are open. Hartmanis [3 ] investigated the lattice of topologies and the lattice of Ti-topologies on a set E. He proved that 2 is complemented if E is finite. If E is finite, A consists of only one element and is trivially complemented. Hartmanis then asked if these lattices are also complemented if E is infinite. It has been shown that 2 is a complemented lattice even when E is infinite, Steiner [4].