Order-compatible topologies on a partially ordered set
E. S. Wolk · Proceedings of the American Mathematical Society · 1958
Introduction.Let X be a partially ordered set (poset) with respect to a relation g, and possessing least and greatest elements 0 and / respectively.There are many known ways of using the order properties of X to define an "intrinsic" topology on X.It is our purpose in this note, instead of considering certain special topologies of this type, to introduce a class of topologies on X which are compatible, in a natural sense, with its order.To this end, let us call a subset S of X up-directed (down-directed) il and only if for all xES and yES there exists zES with z^x,zg^y(z^x, z^y).Also, following