Sum of the spaces on ordered setting

Abstract

Abstract One of the divergences between topology and ordered topology is that some topological concepts such as separation axioms and continuous maps are defined using open neighborhoods or neighborhoods without any difference, however, they are distinct on the ordered topology according to the neighborhoods: Are they open neighborhoods or not? In this paper, we present the concept of sum of the ordered spaces using pairwise disjoint topological ordered spaces and study main properties. Then, we introduce the properties of ordered additive, finitely ordered additive and countably ordered additive which associate topological ordered spaces with their sum. We prove that the properties of being Ti-ordered and strong Ti-ordered spaces are ordered additive, however, the properties of monotonically compact and ordered compact spaces are finitely ordered additive. Also, we define a mapping between two sums of the ordered spaces using mappings between the ordered spaces and deduce some results related to some types of continuity and homeomorphism. We complete this work by determining the conditions under which a topological ordered space is sum of the ordered spaces.