For a long time, the abstract topological structures and concepts seem to be far of applications for most researchers in application fields such as engineers and computer scientists. In addition, it seems that there is a big gap between... more
For a long time, the abstract topological structures and concepts seem to be far of applications for most researchers in application fields such as engineers and computer scientists. In addition, it seems that there is a big gap between these topological structures and real life applications. The main purpose of the present paper is to introduce and study some topological structures induced by binary relations (such as closure operators, and continuous functions) in order to reduce the gap between topologists and who are interested in application of topology. A new method to generate a general topology from the binary relation is constructed. Comparisons between our approaches and the other approaches are investigated. Finally, many examples and counterexamples are introduced.
Violator Spaces were introduced by J. Matoušek et al. in 2008 as generalization of Linear Programming problems. Convex geometries were invented by Edelman and Jamison in 1985 as proper combinatorial abstractions of convexity. Convex... more
Violator Spaces were introduced by J. Matoušek et al. in 2008 as generalization of Linear Programming problems. Convex geometries were invented by Edelman and Jamison in 1985 as proper combinatorial abstractions of convexity. Convex geometries are defined by antiexchange closure operators. We investigate an interrelations between violator spaces and closure spaces and show that violator mapping may be defined by a week version of closure operators. Moreover, we prove that violator spaces with an unique basis satisfies the anti-exchange and the Krein-Milman properties.
Violator Spaces were introduced by J. Matousek et al. in 2008 as generalization of Linear Programming problems. Convex geometries were invented by Edelman and Jamison in 1985 as proper combinatorial abstractions of convexity. Convex... more
Violator Spaces were introduced by J. Matousek et al. in 2008 as generalization of Linear Programming problems. Convex geometries were invented by Edelman and Jamison in 1985 as proper combinatorial abstractions of convexity. Convex geometries are defined by anti-exchange closure operators. We investigate an interrelations between violator spaces and closure spaces and show that violator mapping may be defined by a week version of closure operators. Moreover, we prove that violator spaces with an unique basis satisfies the anti-exchange and the Krein-Milman properties.
