Scott-topology based on transitive binary relation

Document Type : Research Paper


1 Department of Mathematics Faculty of Science, Assiut University, 71516 EGYPT

2 Department of Mathematics, Faculty of Science, New Valley University, EGYPT


In the study of partially ordered sets, topologies such as Scott-topology have shown to be of paramount importance. In order to have analogous topology-like tools in the more general setting of quantitative domains, we introduce a method to construct Scotttopology on a set equipped with a transitive binary relation which we call t-set. As an application of this result there is a Scott-topology associated to any topology induced by its specialization pre-ordered relation. Some relations between this topology and the original topology are investigated.


[1] S. Abramsky, and A. Jung, Domain theory, in S. Abramsky, D. M. Gabbay and T. S. E. Maibaum (eds), Handbook of Logic in Computer Science, 3, Clarendon Press, 1994, pp. 1-168.
[2] G. Bancerek, The way-below Relation, Formalized Mathematics, 6(1) (1997), 169-176. (1997).
[3] G. Birkho , Lattice Theory (3rd ed.). Providence: American Mathematical Society, Col Pub, 1967.
[4] M. Bonsange, F. van Breugal and J. Rutten, Alexandro  and Scott topologies for generalized metric spaces, in : Papers on general topology and its applications, Andima S. et al eds. Eleventh Summer Conference at the University of Southern Main, Annals of the New York Academy of Sciences, 806(1996), 49-68.
[5] R. Engelking, General Topology, Polish Scienti c Publishers, Warszawa, 1977.
[6] G. K. Gierz, K. H. Ho mann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott, A compendium of continuous lattices, Springer-Verlag, Berlin, 1980.
[7] J. Goubault-Larrecq, Non-Hausdor  topology and domain theory, New Mathematical Monographs, vol. 22, Cambridge University Press, Cambridge, 2013.
[8] R. Heckmann, Power domain constructions (Potenzbereich-Konstruktionen), Ph.D. Thesis, Universitat des Saarlandes, December 1999.
[9] J. L. Kelly, General topology, Van Nostrand, New York, 1955.
[10] R. Kummetz, Partially ordered sets with projections and their topology, Ph.D. Thesis, Technische Universitat Dresden, 2000.
[11] K.G. Larsen, G. Winskel, Using information systems to solve recursive domain equations e ectively. In: G. Kahn, D.B. MacQueen and G. Plotkin (eds) Semantics of data types. SDT 1984. Lecture notes in computer science, vol 173. Springer, Berlin, Heidelberg, 1984.
[12] S. Lipschutz, Schaum's outline of theory and problems of general topology, New York : McGraw-Hill INT, 1965.
[13] J. Nino-Salcedo, On continuous posets and their applications, Ph.D. Thesis, Tulane University, 1981.
[14] O. R. Sayed, S. A. Abd El-Baki and N. H. Sayed, Some properties and continuity of transitive binary relational sets, Asia Mathematika, 5(2) (2021), 49-59.
[15] S. Vickers, Information systems for continuous posets, Theoretical Computer Science, 114 (1993), 201-299.
[16] P. Waszkiewicz, Quantitative continuous domains, Ph.D. Thesis, Birmingham University, Edgbaston, B15 2TT, Birmingham, UK, May 2002.
[17] B. Windels, The Scott approach structure: An extension of the Scott topology for quantitative domain theory,Acta Mathematica Academiae Scientiarum Hungaricae, 88(1-2) (2000), 35-44.
[18] Yueli Yue, Wei Yao and Weng Kin Ho, Applications of Scott-closed sets in convex structures, Topology and its Applications, 314 (2022), 108093.
[19] D. Zhao and W.K. Ho, On topologies de ned by irreducible sets, Journal of Logical and Algebraic Methods in Programming, 84 (2015), 185-195.
[20] Z. Zou, Q. Li and W. K. Ho, Domains via approximation operators, Logical Methods in Computer Science, 14(2:6)(2018), 1{17.
  • Receive Date: 09 December 2021
  • Revise Date: 10 April 2022
  • Accept Date: 24 April 2022
  • First Publish Date: 24 April 2022