Crossconvergence: A new bounded topological presentation

• Leseberg, Dieter ^[1] ; Vaziry, Zohreh ^[2]
  1. [1] Free University of Berlin

     Free University of Berlin

     Berlin, Stadt, Alemania

     
  2. [2] Rochester Institute of Technology

     Rochester Institute of Technology

     City of Rochester, Estados Unidos

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 27, Nº. 2, 2026
• Idioma: inglés
• DOI: 10.4995/agt.25481
• Enlaces
  • Texto completo
• Resumen

  • We present crossconvergence, which commonly generalize bornology, preuniform convergence, orderconvergence, b-uniform filtration and grill-determined prenearness. The corresponding construct CCONV forms a strong topological universe, respectively quasitopos, in which quotients are stable under arbitrary products. We also discuss its enlargement to semicrossconvergence and precrossconvergence, respectively.

• Referencias bibliográficas
  • B. Averbukh, On topologies on the underlying set of a topological monoid induced by its unitary extensions, Topol. Algebra Appl. 10, no. 1...
  • M. Baran, S. Kula, T. M. Baran, and M. Quasim, Closure operators in semiuniform convergence spaces, Filomat 30, no. 1 (2016),131-140.https://doi.org/10.2298/FIL1601131B
  • M. Baran, S. Kula and A. Erclyes, T$_0$ and T$_1$ semiuniform convergence spaces, Filomat 27, no. 4 (2013), 537-546.https://doi.org/10.2298/FIL1304537B
  • H. L. Bentley, H. Herrlich and E. Lowen-Colebunders, The category of Cauchy spaces is cartesian closed, Topology Appl. 27 (1987), 105-112.https://doi.org/10.1016/0166-8641(87)90098-8
  • D. Doitchinov, Supertopological spaces and special classes of extensions of topological spaces, In: Faddeev, L. D., Mal'cev, A. A. (eds)...
  • V. A. Efremovic, The geometry of proximity, Mat. Sbornik 32 (73), (1952), 189-200.
  • A. Eriyes, T. Baran, and M. Quasim, Closure operators in constant filter convergence spaces, Konuralp Journal Math. 8, no. 1 (2020), 185-191.
  • B. G. C. Herrejón and F. Mynard, On connected subsets of a convergence space, Int. J. Topol. 2, no. 3 (2025), 13. https://doi.org/10.3390/ijt2030013
  • H. Herrlich, Topology structures, Math. Centre Tracts 52 (1974), 59-122.
  • H. Hogbe-Nlend, Theorié des Bornologies et applications, Lecture Notes in Math. n. 213, Springer Berlin (1971). https://doi.org/10.1007/BFb0069416
  • V. M. Ivanova and A. A. Ivanov, Contiguity spaces and bicompact extensions, DOKL. Akad. Nauk SSSR 127 (1959), 20-22.
  • G. Jäger, Non-symmetric convergence and regularity, Appl. Gen. Topol. 26, no. 1 (2025), 447-467. https://doi.org/10.4995/agt.2025.22432
  • H. J. Kowalski, Topologische Räume, Basel-Stuttgart, Birkhaüser (1961). https://doi.org/10.1007/978-3-0348-6906-5
  • D. Leseberg, A new concept of convergence space, Math. Pann. 19, no. 2 (2008), 291-303.
  • D. Leseberg, On topological induced b-convergences, Topol. Proc. 37 (2011), 293-313.
  • D. Leseberg, Improved nearness research. Adv. Pure Math. 4, no. 11 (2014), 610-626.https://doi.org/10.4236/apm.2014.411070
  • D. Leseberg and Z. Vaziry, The quasitopos of b-uniform filter spaces, Math. Appl. 7 (2018), 155-171. https://doi.org/10.13164/ma.2018.13
  • D. Leseberg and Z. Vaziry, Bounded Topology, LAP, Acad. Pub. (2019).
  • D. Leseberg and Z. Vaziry, Pseudonearness as an important part of bounded topology, ICRAPAM (2021), 258-270.
  • D. Leseberg and Z. Vaziry, The Cauchy completion of symmetric b-uniform filter space, Math. Appl. 10 (2021), 125-142.https://doi.org/10.13164/ma.2021.11142.
  • D. Leseberg and Z. Vaziry, Orderconvergence, a quasitopos for bounded topology BT, Researchgate (2025), 10.13140/RG.2.2.29777.85605.
  • M. W. Lodato, On topological induced generalized proximity relations. I. Proc. Amer. Math. Soc. 15, no. 3 (1964), 417-422. https://doi.org/10.1090/S0002-9939-1964-0161305-3
  • F. Mynard and S. Dolecki, Convergence Foundation of Topology, World Scientific Pub (2016).
  • D. Pallaschke and D. Pumplün, Banach limits revised, Adv. Pure Math. 6, no. 13 (2016), 1022-1036. https://doi.org/10.4236/apm.2016.613075
  • M. J. Penon, Quasitopos, C. R. Acad. Sci. Paris, Sér A. 276 (1973), 237-240.
  • G. Preuss, Semiuniform convergence spaces, Math. Jap. 41(1995), 465-491.
  • G. Preuss, Foundation of Topology. An approach to Convenient Topology, Dordrecht, Kluwer Acad. Pub. (2002).https://doi.org/10.1007/978-94-010-0489-3
  • G. Preuss, Prefilter spaces and a precompletion of preuniform convergence spaces related to some well-known completions, Topology Appl. 156,...
  • D. Pumplün, A universal compactification of topological positively convex sets and moduls, Journal of Convex Analysis 41 (2011), 255-267.
  • Y. M. Smirnov, On completeness of proximity spaces, Dokl. Acad. Nauk SSSR 88 (1953), 761-764.
  • A. Weil, Sur les espaces à structures uniformes et sur la toplogie générale, Act. Sci. Ind. 551 (1937).
  • O. Wyler, On convergence of filters and ultrafilters to subsets, Lect. Notes Comp. Sci. 393 (1988), 340-350. https://doi.org/10.1007/3-540-51722-7_23