The Upper Interval Topology, Property M and Compactness

A topology on a lattice L or more generally on a partially ordered set P is called an intrinsic topology if it is de ned directly from the order Early ex amples of such topologies were the interval topology of O Frink Fr and the order topology introduced by G Birkho Bi The early intrinsic topolo gies were typically symmetric i e the topology de ned on L agreed with the topology de ned on L the dual lattice with the order reversed The theory of continuous lattices however provided strong motivation for the consideration of topologies such as the Scott topology or the hull kernel topology which were not symmetric in this sense COMP indeed were not even T Indeed from hindsight it is very natural to consider intrinsic topologies that are not symmetric Given any T topology on a set there results a partial order the order of specialization de ned by x y if and only if x fyg It is then natural to consider order consistent topologies on a partially ordered set P topologies for which the order of specialization agrees with the original given partial order These typically satisfy the T separation axiom but nothing stronger There is a weakest order consistent topology on a partially ordered set which we call the lower interval topology and which has as a subbasis for the closed sets all principal ideals x x P where x fy y xg This topology was called the upper topology in COMP and has also been called the weak topology We quickly recall basic notions of ordered sets arising in continuous domain theory see for example AJ or COMP Let P be a partially ordered set or poset A non empty subset D of P is directed if x y D implies there exists z D with x z and y z A set A is a lower set if