Quasi-uniform completions of partially ordered spaces

In this paper we define partially ordered quasi-uniform spaces (X, $$\mathfrak{U}$$ , ≤) (PO-quasi-uniform spaces) as those space with a biconvex quasi-uniformity $$\mathfrak{U}$$ on the poset (X, ≤) and give a construction of a (transitive) biconvex compatible quasi-uniformity on a partially ordered topological space when its topology satisfies certain natural conditions. We also show that under certain conditions on the topology $$\tau _{\mathfrak{U}*} $$ of a PO-quasi-uniform space (X, $$\mathfrak{U}$$ , ≤), the bicompletion $$(\tilde X,\tilde {\mathfrak{U}})$$ of (X, $$\mathfrak{U}$$ ) is also a PO-quasi-uniform space ( $$(\tilde X,\tilde {\mathfrak{U}})$$ , ⪯) with a partial order ⪯ on $$\tilde X$$ that extends ≤ in a natural way.

Ordered compactifications with countable remainders

It is shown that if a partially-ordered topological space X admits a finite-point T2-ordered compactification, then it admits a countable T2-ordered compactification if and only if it admits n−point T2-ordered compactifications for all n beyond some integer m.

Posets of ordered compactifications

If (X′, τ′, ≤′) is an ordered compactification of the partially ordered topological space (X, τ, ≤) such that ≤′ is the smallest order that renders (X′, τ′, ≤′) a T2-ordered compactification of X, then X′ is called a Nachbin (or order-strict) compactification of (X, τ, ≤). If (X′, τ′, ≤*) is a finite-point ordered compactification of (X, τ, ≤), the Nachbin order ≤′ for (X′, τ′) is described in terms of (X, τ, ≤) and X′. When given the usual order relation between compactifications (ordered compactifications), posets of finite-point Nachbin compactifications are shown to have the same order structure as the poset of underlying topological compactifications. Though posets of arbitrary finite-point ordered compactifications are shown to be less well behaved, conditions for their good behavior are studied. These results are used to examine the lattice structure of the set of all ordered compactifications of the ordered topological space (X, τ, ≤).