Mathematica Bohemica, Vol. 142, No. 3, pp. 277-298, 2017


Probabilistic approach spaces

Gunther Jäger

Received October 22, 2015.  First published January 3, 2017.

Abstract:  We study a probabilistic generalization of Lowen's approach spaces. Such a probabilistic approach space is defined in terms of a probabilistic distance which assigns to a point and a subset a distance distribution function. We give a suitable axiom scheme and show that the resulting category is isomorphic to the category of left-continuous probabilistic topological convergence spaces and hence is a topological category. We further show that the category of Lowen's approach spaces is isomorphic to a simultaneously bireflective and bicoreflective subcategory and that the category of probabilistic quasi-metric spaces is isomorphic to a bicoreflective subcategory of the category of probabilistic approach spaces.
Keywords:  approach space; probabilistic approach space; probabilistic convergence space; probabilistic metric space
Classification MSC:  54A20, 54E70, 54E99
DOI:  10.21136/MB.2017.0064-15


References:
[1] S. Abramsky, A. Jung: Domain Theory. Handbook of Logic and Computer Science. Vol. 3 (S. Abramsky et al., eds.). Claredon Press, Oxford University Press, New York (1994), 1-168. MR 1365749
  [2] J. Adámek, H. Herrlich, G. E. Strecker: Abstract and Concrete Categories - The Joy of Cats. John Wiley & Sons, New York (1990). MR 1051419 | Zbl 0695.18001
  [3] P. Brock: Probabilistic convergence spaces and generalized metric spaces. Int. J. Math. Math. Sci. 21 (1998), 439-452. DOI 10.1155/S0161171298000611 | MR 1620306 | Zbl 0905.54005
  [4] P. Brock, D. C. Kent: Approach spaces, limit tower spaces, and probabilistic convergence spaces. Appl. Categ. Struct. 5 (1997), 99-110. DOI 10.1023/A:1008633124960 | MR 1456517 | Zbl 0885.54008
  [5] R. C. Flagg: Quantales and continuity spaces. Algebra Univers. 37 (1997), 257-276. DOI 10.1007/s000120050018 | MR 1452402 | Zbl 0906.54024
  [6] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D.S. Scott: Continuous Lattices and Domains. Encyclopedia of Mathematics and Its Applications 93. Cambridge University Press, Cambridge (2003). DOI 10.1017/CBO9780511542725 | MR 1975381 | Zbl 1088.06001
  [7] G. Jäger: A convergence theory for probabilistic metric spaces. Quaestiones Math. 38 (2015), 587-599. DOI 10.2989/16073606.2014.981734 | MR 3403669
  [8] H.-J. Kowalsky: Limesräume und Komplettierung. Math. Nachr. 12 (1954), 301-340. DOI 10.1002/mana.19540120504 | MR 0073147 | Zbl 0056.41403
  [9] R. Lowen: Approach spaces: A common supercategory of TOP and MET. Math. Nachr. 141 (1989), 183-226. DOI 10.1002/mana.19891410120 | MR 1014427 | Zbl 0676.54012
  [10] R. Lowen: Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad. Oxford Mathematical Monographs. Clarendon Press, Oxford (1997). MR 1472024 | Zbl 0891.54001
  [11] R. Lowen: Index Analysis: Approach Theory at Work. Springer Monographs in Mathematics. Springer, London (2015). DOI 10.1007/978-1-4471-6485-2 | MR 3308451 | Zbl 1311.54002
  [12] K. Menger: Statistical metrics. Proc. Natl. Acad. Sci. USA 28 (1942), 535-537. DOI 10.1073/pnas.28.12.535 | MR 0007576 | Zbl 0063.03886
  [13] G. Preuss: Foundations of Topology: An Approach to Convenient Topology. Kluwer Academic Publishers, Dordrecht (2002). DOI 10.1007/978-94-010-0489-3 | MR 2033142 | Zbl 1058.54001
  [14] G. D. Richardson, D. C. Kent: Probabilistic convergence spaces. J. Aust. Math. Soc., Ser. A 61 (1996), 400-420. DOI 10.1017/S1446788700000483 | MR 1420347 | Zbl 0943.54002
  [15] A.-F. Roldán-López-de-Hierro, M. de la Sen, J. Martínez-Moreno, C. Roldán-López-de-Hierro: An approach version of fuzzy metric spaces including an ad hoc fixed point theorem. Fixed Point Theory Appl. (electronic only) (2015), Article ID 33, 23 pages. DOI 10.1186/s13663-015-0276-7 | MR 3316772 | Zbl 1310.54004
  [16] S. Saminger-Platz, C. Sempi: A primer on triangle functions I. Aequationes Math. 76 (2008), 201-240. DOI 10.1007/s00010-008-2936-8 | MR 2461890 | Zbl 1162.54013
  [17] S. Saminger-Platz, C. Sempi: A primer on triangle functions II. Aequationes Math. 80 (2010), 239-268. DOI 10.1007/s00010-010-0038-x | MR 2739176 | Zbl 1213.39031
  [18] B. Schweizer, A. Sklar: Statistical metric spaces. Pac. J. Math. 10 (1960), 313-334. DOI 10.2140/pjm.1960.10.313 | MR 0115153 | Zbl 0091.29801
  [19] B. Schweizer, A. Sklar: Probabilistic metric spaces. North Holland Series in Probability and Applied Mathematics. North Holland, New York (1983). MR 0790314 | Zbl 0546.60010
  [20] R. M. Tardiff: Topologies for probabilistic metric spaces. Pac. J. Math. 65 (1976), 233-251. DOI 10.2140/pjm.1976.65.233 | MR 0423315 | Zbl 0337.54004
  [21] A. Wald: On a statistical generalization of metric spaces. Proc. Natl. Acad. Sci. USA 29 (1943), 196-197. DOI 10.1073/pnas.29.6.196 | MR 0007950 | Zbl 0063.08119

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Affiliations:   Gunther Jäger, School of Mechanical Engineering, University of Applied Sciences Stralsund, Zur Schwedenschanze 15, 18435 Stralsund, Germany, e-mail: gunther.jaeger@fh-stralsund.de

 
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