Advanced search
1 file | 411.04 KB

Topologies induced by the representation of a betweenness relation as a family of order relations

Huapeng Zhang (UGent) , Raul Perez Fernandez (UGent) and Bernard De Baets (UGent)
Author
Organization
Abstract
Betweenness relations are the mathematical formalization of the geometrical notion of an element being in between other two elements. In this paper, we exploit a well-known result representing a betweenness relation as a family of order relations and analyse the corresponding family of induced (Alexandrov) topologies. In particular, the intersection of this family of topologies is proved to be the anti-discrete topology and a necessary and sufficient condition for the supremum of this family of topologies to be the discrete topology is provided. Interestingly, this condition is proved to hold when dealing with a finite set. We end with a discussion on the relation between the topology induced by an order relation or a metric and the family of topologies induced by the betweenness relation induced by the same order relation or metric.
Keywords
Geometry and Topology

Downloads

  • KERMIT-A1-523.pdf
    • full text
    • |
    • open access
    • |
    • PDF
    • |
    • 411.04 KB

Citation

Please use this url to cite or link to this publication:

Chicago
Zhang, Huapeng, Raul Perez Fernandez, and Bernard De Baets. 2019. “Topologies Induced by the Representation of a Betweenness Relation as a Family of Order Relations.” Topology and Its Applications 258: 100–114.
APA
Zhang, Huapeng, Perez Fernandez, R., & De Baets, B. (2019). Topologies induced by the representation of a betweenness relation as a family of order relations. Topology and its Applications, 258, 100–114.
Vancouver
1.
Zhang H, Perez Fernandez R, De Baets B. Topologies induced by the representation of a betweenness relation as a family of order relations. Topology and its Applications. 2019;258:100–14.
MLA
Zhang, Huapeng, Raul Perez Fernandez, and Bernard De Baets. “Topologies Induced by the Representation of a Betweenness Relation as a Family of Order Relations.” Topology and its Applications 258 (2019): 100–114. Print.
@article{8626845,
  abstract     = {Betweenness relations are the mathematical formalization of the geometrical notion of an element being in between other two elements. In this paper, we exploit a well-known result representing a betweenness relation as a family of order relations and analyse the corresponding family of induced (Alexandrov) topologies. In particular, the intersection of this family of topologies is proved to be the anti-discrete topology and a necessary and sufficient condition for the supremum of this family of topologies to be the discrete topology is provided. Interestingly, this condition is proved to hold when dealing with a finite set. We end with a discussion on the relation between the topology induced by an order relation or a metric and the family of topologies induced by the betweenness relation induced by the same order relation or metric.},
  author       = {Zhang, Huapeng and Perez Fernandez, Raul and De Baets, Bernard},
  issn         = {0166-8641},
  journal      = {Topology and its Applications},
  keywords     = {Geometry and Topology},
  language     = {eng},
  pages        = {100--114},
  title        = {Topologies induced by the representation of a betweenness relation as a family of order relations},
  url          = {http://dx.doi.org/10.1016/j.topol.2019.02.045},
  volume       = {258},
  year         = {2019},
}

Altmetric
View in Altmetric