Probabilistic approach spaces

Gunther Jäger

Mathematica Bohemica (2017)

  • Volume: 142, Issue: 3, page 277-298
  • ISSN: 0862-7959

Abstract

top
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.

How to cite

top

Jäger, Gunther. "Probabilistic approach spaces." Mathematica Bohemica 142.3 (2017): 277-298. <http://eudml.org/doc/294141>.

@article{Jäger2017,
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.},
author = {Jäger, Gunther},
journal = {Mathematica Bohemica},
keywords = {approach space; probabilistic approach space; probabilistic convergence space; probabilistic metric space},
language = {eng},
number = {3},
pages = {277-298},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Probabilistic approach spaces},
url = {http://eudml.org/doc/294141},
volume = {142},
year = {2017},
}

TY - JOUR
AU - Jäger, Gunther
TI - Probabilistic approach spaces
JO - Mathematica Bohemica
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 142
IS - 3
SP - 277
EP - 298
AB - 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.
LA - eng
KW - approach space; probabilistic approach space; probabilistic convergence space; probabilistic metric space
UR - http://eudml.org/doc/294141
ER -

References

top
  1. Abramsky, S., Jung, A., 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. (1994) MR1365749
  2. Adámek, J., Herrlich, H., Strecker, G. E., Abstract and Concrete Categories---The Joy of Cats, John Wiley & Sons, New York (1990). (1990) Zbl0695.18001MR1051419
  3. Brock, P., 10.1155/S0161171298000611, Int. J. Math. Math. Sci. 21 (1998), 439-452. (1998) Zbl0905.54005MR1620306DOI10.1155/S0161171298000611
  4. Brock, P., Kent, D. C., 10.1023/A:1008633124960, Appl. Categ. Struct. 5 (1997), 99-110. (1997) Zbl0885.54008MR1456517DOI10.1023/A:1008633124960
  5. Flagg, R. C., 10.1007/s000120050018, Algebra Univers. 37 (1997), 257-276. (1997) Zbl0906.54024MR1452402DOI10.1007/s000120050018
  6. Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. W., Scott, D.S., 10.1017/CBO9780511542725, Encyclopedia of Mathematics and Its Applications 93. Cambridge University Press, Cambridge (2003). (2003) Zbl1088.06001MR1975381DOI10.1017/CBO9780511542725
  7. Jäger, G., 10.2989/16073606.2014.981734, Quaestiones Math. 38 (2015), 587-599. (2015) MR3403669DOI10.2989/16073606.2014.981734
  8. Kowalsky, H.-J., 10.1002/mana.19540120504, Math. Nachr. 12 (1954), 301-340. (1954) Zbl0056.41403MR0073147DOI10.1002/mana.19540120504
  9. Lowen, R., 10.1002/mana.19891410120, Math. Nachr. 141 (1989), 183-226. (1989) Zbl0676.54012MR1014427DOI10.1002/mana.19891410120
  10. Lowen, R., Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs. Clarendon Press, Oxford (1997). (1997) Zbl0891.54001MR1472024
  11. Lowen, R., 10.1007/978-1-4471-6485-2, Springer Monographs in Mathematics. Springer, London (2015). (2015) Zbl1311.54002MR3308451DOI10.1007/978-1-4471-6485-2
  12. Menger, K., 10.1073/pnas.28.12.535, Proc. Natl. Acad. Sci. USA 28 (1942), 535-537. (1942) Zbl0063.03886MR0007576DOI10.1073/pnas.28.12.535
  13. Preuss, G., 10.1007/978-94-010-0489-3, Kluwer Academic Publishers, Dordrecht (2002). (2002) Zbl1058.54001MR2033142DOI10.1007/978-94-010-0489-3
  14. Richardson, G. D., Kent, D. C., 10.1017/S1446788700000483, J. Aust. Math. Soc., Ser. A 61 (1996), 400-420. (1996) Zbl0943.54002MR1420347DOI10.1017/S1446788700000483
  15. Roldán-López-de-Hierro, A.-F., Sen, M. de la, Martínez-Moreno, J., Hierro, C. Roldán-López-de-, 10.1186/s13663-015-0276-7, Fixed Point Theory Appl. (electronic only) (2015), Article ID 33, 23 pages. (2015) Zbl1310.54004MR3316772DOI10.1186/s13663-015-0276-7
  16. Saminger-Platz, S., Sempi, C., 10.1007/s00010-008-2936-8, Aequationes Math. 76 (2008), 201-240. (2008) Zbl1162.54013MR2461890DOI10.1007/s00010-008-2936-8
  17. Saminger-Platz, S., Sempi, C., 10.1007/s00010-010-0038-x, Aequationes Math. 80 (2010), 239-268. (2010) Zbl1213.39031MR2739176DOI10.1007/s00010-010-0038-x
  18. Schweizer, B., Sklar, A., 10.2140/pjm.1960.10.313, Pac. J. Math. 10 (1960), 313-334. (1960) Zbl0091.29801MR0115153DOI10.2140/pjm.1960.10.313
  19. Schweizer, B., Sklar, A., nema, North Holland Series in Probability and Applied Mathematics. North Holland, New York (1983). (1983) Zbl0546.60010MR0790314DOInema
  20. Tardiff, R. M., 10.2140/pjm.1976.65.233, Pac. J. Math. 65 (1976), 233-251. (1976) Zbl0337.54004MR0423315DOI10.2140/pjm.1976.65.233
  21. Wald, A., 10.1073/pnas.29.6.196, Proc. Natl. Acad. Sci. USA 29 (1943), 196-197. (1943) Zbl0063.08119MR0007950DOI10.1073/pnas.29.6.196

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.