<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>T</italic><italic>i</italic>-ordered and strong <italic>T</italic><italic>i</italic>-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.</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.

Sum of the spaces on ordered setting

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Sum of the spaces on ordered setting"}

One of the divergences between topology and ordered topology is that some topological concepts such as separation axioms and continuous maps are defined using...

Sum of the spaces on ordered setting

Published Online: Oct 02, 2020

Page range: 255 - 265

Received: Jun 10, 2020

Accepted: Sep 14, 2020

DOI: https://doi.org/10.2478/mjpaa-2020-0020

Keywords
Sum of the ordered spaces, Additive ordered property, -ordered space, Monotonically compact space, Continuity

© 2020 T. M. Al-shami, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Sum of the spaces on ordered setting

Published Online: Oct 02, 2020

Page range: 255 - 265

Received: Jun 10, 2020

Accepted: Sep 14, 2020

DOI: https://doi.org/10.2478/mjpaa-2020-0020

KeywordsSum of the ordered spaces, Additive ordered property, -ordered space, Monotonically compact space, Continuity

© 2020 T. M. Al-shami, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Sum of the ordered spaces, Additive ordered property, -ordered space, Monotonically compact space, Continuity