On Partially Ordered Sets Possessing a Unique Order-compatible Topology

1. Introduction. Let A" be a partially ordered set (poset) with respect to a relation ^, and possessing least and greatest elements O and / respectively. Let us call a subset S of A" up-directed (down-directed) if and only if for all xES and y ES there exists zGS such that 3^x, z^y (z^x, z^y). Following McShane [2], we call a subset K of X Dedekind-closed if and only if whenever S is an up-directed subset of K and y = l.u.b. (S), or S is a down-directed subset of K and y = g.l.b. (S), we have yEK. Let £> denote the topology on X whose closed sets are the Dedekind-closed subsets of X. Let d denote the well-known interval topology on X, which is obtained by taking all sets of the form [a, b]= {xEX\a^x^b} as a sub-base for the closed sets. Continuing an investigation which was begun in [5], we shall call a topology 3 on X order-compatible if and only if â ^ 3 ^ 3D. X is said to have a unique order-compatible topology if and only if its é and 3) topologies are identical. In [5] we obtained a simple sufficient condition for a poset to possess a unique order-compatible topology. This result has recently been strengthened by Naito [3]. Let us call a subset K of X diverse if and only if xEK, yEK, and x^y imply x<y. Naito has shown that if a poset X contains no infinite diverse subset, then it possesses a unique order-compati