A compact topology for a lattice

Abstract

Introduction. In this paper we shall study a compact intrinsic topology for a lattice and obtain a few relationships between this topology and certain well-known intrinsic topologies for lattices. We obtain as a result the fact that for a large class of lattices, compactness of the order topology implies that our compact topology and the order topology coincide. Let L be a lattice and { xa }, a net in L. We define the limit inferior, L* {xa } = VaAbla Xb, and the limit superior, L* {xa} = AaVb?a Xb. Then, provided they exist, L* { xa} I L* {Ixa }. If L* { xa I = L* { xa } = x, we say that the net { xa } order converges to x. Let C be a subset of L. C is said to be order closed iff no net in C order converges to a point outside of C. The collection of order closed sets comprises the closed sets for a topology for L. We call this topology the order topology for L and designate it by O(L). The collection of sets of the form {x:x;c} and {x:x>c} for cGL forms the sub-base of the closed sets of a weaker topology called the interval topology and designated by I(L). It is known that for any lattice L, L is complete iff L with the interval topology is compact. (See G. Birkhoff's Lattice theory.)