Compactification of closed preordered spaces

Abstract

A topological preordered space admits a Hausdorff T2-preorder compactification if and only if it is Tychonoff and the preorder is represented by the family of continuous isotone functions. We construct the largest Hausdorff T2-preorder compactification for these spaces and clarify its relation with Nachbin's compactification. Under local compactness the problem of the existence and identification of the smallest Hausdorff T2-preorder compactification is considered.2010 MSC: 54E15 (primary), 54F05, 54E55, 06F30 (secondary).

Causality in noncommutative two-sheeted space-times

“…Also, the role of the unitisation is only technical as we are eventually interested in the causal relation on M(A), which we regard as the space of physical states. On the other hand, in the commutative case the choice of the unitisation is equivalent to picking a suitable compactification of the space-time M. The latter is directly related to an old-standing problem of attaching a 'boundary' to a, possibly singular, space-time and extending the causal relation to it [3].…”

“…The latter has very deep consequences for physical models as it sets fundamental restrictions on the evolution of physical processes. On the mathematical side, the causal structure on a Lorentzian manifold M induces a partial order relation on the set of points of M. The properties of this order have been studied by several authors (see for instance [1,2,3,4]).…”

Convexity and quasi-uniformizability of closed preordered spaces

“…Quasi-uniformizable topological preordered spaces are among the most well behaved topological preordered spaces. They admit completions and compactifications [7,8,45,27], and under second countability they can be shown to be quasi-pseudo-metrizable [35].…”

“…Contrary to what happens in the usual discrete-preorder case, normally preordered spaces need not be completely regularly preordered spaces (see example 1.5), nevertheless the preorder analog of Urysohn's separation lemma implies that convex normally preordered spaces are completely regularly preordered spaces. Completely regularly ordered spaces admit the Nachbin's T 2 -ordered compactification nE (see [8] and [27] for the preorder case).…”

Isotone extension of mappings

“…Другая типичная ситуация это продолжение изотонного непрерывного отображения, на упорядоченном или предупорядоченном топологическом пространстве до изотонного непрерывного отображения на компактификации этого пространства (см., например, [2] и приведенные там ссылки). Как отмечается в [2], такого рода исследования мотивированы, в частности, попытками перенести причинно-следственные отношения (casual relations) на идеальные границы лоренцевых многообразий.…”