The interval topology of a lattice
E. S. Northam · Proceedings of the American Mathematical Society · 1953
In this paper we obtain a necessary condition that a lattice be Hausdorff in its interval topology.This condition, stated in Proposition 2, can be applied to show that the interval topology of a Boolean algebra is Hausdorff if and only if every element is over an atom and that an Z-group need not be Hausdorff in its interval topology.The former supplies a more or less complete answer to problem 76 of Birkhoff [l] and the latter solves 104.In addition a necessary and sufficient condition is obtained for a point to be isolated in the interval topology, thus answering, in part, problem 21.Frink [2] has defined the interval topology of a lattice (or partly ordered set) by taking as a sub-basis for the closed sets all finite [a, b] and infinite [-«>, a], [a, ] closed intervals.A basis for the closed sets is then the collection of all finite unions of such intervals.As usual we say that a space is Hausdorff if for any two distinct points x and y, there exist disjoint open sets U and V with #£ U and y£ V. Furthermore it is easily seen that we may select U and V from any given basis of open sets.Looking at the complements of U and V we obtain the dual requirement that given any two distinct points, the space can be covered by two closed sets each of which contains exactly one of the points, and in addition we may select these sets from any given basis for the closed sets.In particular: