In domain theory, by a poset model of a T1 topological space X we usually
mean a poset P such that the subspace Max(P) of the Scott space of P
consisting of all maximal points is homeomorphic to X. The poset models of
T1 spaces have been extensively studied by many authors. In this paper we
investigate another type of poset models: lower topology models. The lower
topology ?(P) on a poset P is one of the fundamental intrinsic topologies on
the poset, which is generated by the sets of the form P\?x, x ? P. A lower
topology poset model (poset LT-model) of a topological space X is a poset P
such that the space Max?(P) of maximal points of P equipped with the
relative lower topology is homeomorphic to X. The studies of such new models
reveal more links between general T1 spaces and order structures. The main
results proved in this paper include (i) a T1 space is compact if and only
if it has a bounded complete algebraic dcpo LT-model; (ii) a T1 space is
second-countable if and only if it has an ?-algebraic poset LT-model; (iii)
every T1 space has an algebraic dcpo LT-model; (iv) the category of all T1
space is equivalent to a category of bounded complete posets. We will also
prove some new results on the lower topology of different types of posets.
Maximal point spaces of dcpos satisfying the Lawson condition
