Quantale-valued uniform convergence towers for quantale-valued metric spaces

Abstract

We show that quantale-valued metric spaces and quantale-valued partial metric spaces allow natural quantale-valued uniform convergence structures. We furthermore characterize quantale-valued metric spaces and quantale-valued partial metric spaces by these quantale-valued uniform convergence structures. For special choices of the quantale, the results specialize to metric spaces and probabilistic metric spaces.

Keywords

• $L$-metric space,$L$-partial metric space,$L$-uniform convergence tower space

References

1. [1] J. Adámek, H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1989.
2. [2] T.M.G. Ahsanullah and G. Jäger, Probabilistic uniform convergence spaces redefined, Acta Math. Hungarica, 146, 376–390, 2015.
3. [3] N. Bourbaki, General topology, Chapters 1 – 4, Springer Verlag, Berlin - Heidelberg - New York - London - Paris - Tokyo, 1990.
4. [4] P. Brock, Probabilistic convergence spaces and generalized metric spaces, Int. J. Math. and Math. Sci. 21, 439–452, 1998.
5. [5] P. Brock and D.C. Kent, Approach spaces, limit tower spaces, and probabilistic con- vergence spaces, Appl. Cat. Structures, 5, 99–110, 1997.
6. [6] R.C. Flagg, Completeness in continuity spaces, in: Category Theory 1991, CMS Conf. Proc. 13, 183–199, 1992.
7. [7] R.C. Flagg, Quantales and continuity spaces, Algebra Univers. 37, 257–276, 1997.
8. [8] L.C. Florescu, Probabilistic convergence structures, Aequationes Math. 38, 123–145, 1989.

9. [9] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott, Continuous lattices and domains, Cambridge University Press, 2003.
10. [10] G. Jäger, A convergence theory for probabilistic metric spaces, Quaest. Math. 38, 587–599, 2015.
11. [11] G. Jäger, A common framework for lattice-valued, probabilistic and approach uniform (convergence) spaces, Iran. J. Fuzzy Syst. 14, 67–82, 2017.
12. [12] G. Jäger and T.M.G. Ahsanullah, Characterization of quantale-valued metric spaces and quantale-valued partial metric spaces by convergence, Appl. Gen. Topol. 19 (1), 129–144, 2018.
13. [13] R. Kopperman, S. Matthews and H. Pajoohesh, Partial metrizability in value quan- tales, Appl. Gen. Topol. 5, 115 – 127, 2004.
14. [14] F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rendiconti del Seminario Matematico e Fisico di Milano, 43, 135–166, 1973. Reprinted in: Repr. Theory Appl. Categ. 1, 1–37, 2002.
15. [15] Y.J. Lee and B. Windels, Transitivity in uniform approach theory, Int. J. Math. Math. Sci. 32, 707–720, 2002.
16. [16] R. Lowen, Approach spaces. The missing link in the topology-uniformity-metric triad, Claredon Press, Oxford, 1997.
17. [17] H. Nusser, A generalization of probabilistic uniform spaces, Appl. Categ. Structures, 10, 81–98, 2002.
18. [18] G. Preuss, Foundations of topology. An approach to convenient topology, Kluwer Aca- demic Publishers, Dordrecht, 2002.
19. [19] G.N. Raney, A subdirect-union representation for completely distributive complete lattices, Proc. Amer. Math. Soc. 4, 518–512, 1953.
20. [20] G.D. Richardson and D.C. Kent, Probabilistic convergence spaces, J. Austral. Math. Soc. (Series A), 61, 400–420, 1996.
21. [21] B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland, New York, 1983.
22. [22] R.M. Tardiff, Topologies for probabilistic metric spaces, Pacific J. Math. 65, 233–251, 1976.