Convexity and quasi-uniformizability of closed preordered spaces

In many applications it is important to establish if a given topological preordered space has a topology and a preorder which can be recovered from the set of continuous isotone functions. Under antisymmetry this property, also known as quasi-uniformizability, allows one to compactify the topological space and to extend its order dynamics. In this work we study locally compact $\sigma$-compact spaces endowed with a closed preorder. They are known to be normally preordered, and it is proved here that if they are locally convex, then they are convex, in the sense that the upper and lower topologies generate the topology. As a consequence, under local convexity they are quasi-uniformizable. The problem of establishing local convexity under antisymmetry is studied. It is proved that local convexity holds provided the convex hull of any compact set is compact. Furthermore, it is proved that local convexity holds whenever the preorder is compactly generated, a case which includes most examples of interest, including preorders determined by cone structures over differentiable manifolds. The work ends with some results on the problem of quasi-pseudo-metrizability. As an application, it is shown that every stably causal spacetime is quasi-uniformizable and every globally hyperbolic spacetime is strictly quasi-pseudo-metrizable.

💡 Research Summary

The paper investigates the interplay between topology and preorder in topological preordered spaces (TPS), focusing on conditions that guarantee the space can be recovered from its continuous isotone functions—a property known as quasi‑uniformizability. The setting is a locally compact, σ‑compact Hausdorff space X equipped with a closed preorder R (i.e., R is a closed subset of X×X). The authors first recall that under these hypotheses the space is normally preordered: upper and lower open sets separate points and closed decreasing/increasing sets, a classical result that ensures a good separation between order and topology.

The central contribution is the proof that local convexity implies global convexity for such spaces. Local convexity means that for every point x there exists an open neighbourhood U such that the intersection of U with any upper (resp. lower) set containing x is still an upper (resp. lower) set in U. When this holds, the upper topology τ⁺ and the lower topology τ⁻ together generate the original topology τ, i.e., τ = τ⁺ ∨ τ⁻. Consequently the space becomes convex in the order‑topological sense, and convex normally preordered spaces are known to be quasi‑uniformizable. Hence, under local convexity the TPS admits a quasi‑uniformity whose associated continuous isotone functions separate points and recover both τ and R.

The authors then address how to guarantee local convexity. Two sufficient conditions are presented. First, if the convex hull of every compact subset of X is compact, then local convexity follows. This condition is automatically satisfied when the preorder is compactly generated: there exists a compact set C⊂X×X such that R is the smallest closed preorder containing C. In this case any pair (x,y)∈R can be linked by a finite C‑chain, and the compactness of C forces the convex hull of compact sets to remain compact, yielding local convexity.

A major class of examples fitting the compactly generated hypothesis consists of cone‑induced preorders on differentiable manifolds. At each point p of a manifold M a closed convex cone Cₚ⊂TₚM is prescribed, varying continuously with p, and the preorder x≤y is defined by the existence of a piecewise‑smooth curve whose tangent vectors lie in the corresponding cones. Because each cone is closed and bounded in finite dimensions, the induced preorder is compactly generated, and therefore the associated TPS is locally convex, convex, and quasi‑uniformizable.

The paper also explores the quasi‑pseudo‑metrization problem. For convex normally preordered spaces one can construct a pair of quasi‑pseudo‑metrics (d⁺, d⁻) that generate τ⁺ and τ⁻ respectively. When the underlying spacetime is globally hyperbolic, these metrics can be chosen to be complete and to reflect the causal order precisely, leading to strict quasi‑pseudo‑metrizability. As an application, the authors prove that every stably causal spacetime (a spacetime whose causal order is antisymmetric and admits a time function) is quasi‑uniformizable, and every globally hyperbolic spacetime is strictly quasi‑pseudo‑metrizable.

In summary, the work establishes a clear hierarchy of conditions: closed preorder + local compactness + σ‑compactness ⇒ normal preorder; add local convexity ⇒ convex ⇒ quasi‑uniformizable; ensure compactly generated preorder (or compact convex hulls of compact sets) ⇒ local convexity automatically. These results unify several strands of order‑topology, provide new tools for the causal analysis of Lorentzian manifolds, and open the way to compactifications and metric‑type representations of a broad class of ordered topological spaces.