Year 2019, Volume 48 , Issue 6, Pages 1653 - 1666 2019-12-08

Bornological quasi-metrizability in generalized topology

Artur PİĘKOSZ [1] , Eliza WAJCH [2]


A concept of quasi-metrizability with respect to a bornology of a generalized topological space in the sense of Delfs and Knebusch is introduced. Quasi-metrization theorems for generalized bornological universes are deduced. A uniform quasi-metrizability with respect to a bornology is studied. The class of locally small spaces is considered and a possibly larger class of weakly locally small spaces is defined. The proofs and numerous examples are given in ZF. An example of a weakly locally small space which is not locally small is constructed under ZF+CC. Several categories, relevant to generalized bornological universes, are defined and shown to be topological constructs.
Delfs-Knebusch generalized topological space, quasi-metric, bornology, topological category, ZF
  • [1] J. Adámek, H. Herrlich and G.E. Strecker, Abstract and Concrete Categories, the Joy of Cats, Wiley, New York, 1990.
  • [2] A. Appert and K. Fan, Espaces Topologiques Intermédiaires, Act. Sci. et Ind. 1121, Hermann, Paris, 1951.
  • [3] E. Colebunders and R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (5), 1547–1556, 2004.
  • [4] Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar. 96 (4), 351–357, 2002.
  • [5] H. Delfs and M. Knebusch, Locally Semialgebraic Spaces, in: Lecture Notes in Math. 1173, Springer, Berlin-Heidelberg, 1985.
  • [6] M.J. Edmundo and L. Prelli, Sheaves on T-topologies, J. Math. Soc. Japan, 68 (1), 347–381, 2016.
  • [7] P. Fletcher and W.F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982.
  • [8] I. Garrido and A.S. Meroño, Uniformly metrizable bornologies, J. Convex Anal. 20 (1), 285–299, 2013.
  • [9] H. Herrlich, Axiom of Choice, Springer-Verlag, Berlin-Heidelberg, 2006.
  • [10] D. Hofmann, G.J. Seal and W. Tholen, Monoidal Topology, A Categorical Approach to Order, Metric, and Topology, Cambridge University Press, Cambridge, 2014.
  • [11] H. Hogbe-Nlend, Bornologies and Functional Analysis, North-Holland, Amsterdam, 1977.
  • [12] S.T. Hu, Boundedness in a topological space, J. Math. Pures Appl. 28, 287–320, 1949.
  • [13] J.C. Kelly, Bitopological spaces, Proc. Lond. Math. Soc. 13, 71–89, 1963.
  • [14] M. Knebusch, Weakly Semialgebraic Spaces, in: Lecture Notes in Math. 1367, Springer-Verlag, New York, 1989.
  • [15] K. Kunen, The Foundations of Mathematics, Individual Authors and College Publications, London, 2009.
  • [16] S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, Berlin- Heidelberg, 1971.
  • [17] A. Piękosz, On generalized topological spaces I, Ann. Polon. Math. 107 (3), 217–241, 2013.
  • [18] A. Piękosz, On generalized topological spaces II, Ann. Polon. Math. 108 (2), 185–214, 2013.
  • [19] A. Piękosz, O-minimal homotopy and generalized (co)homology, Rocky Mountain J. Math. 43 (2), 573–617, 2013.
  • [20] A. Piękosz and E. Wajch, Compactness and compactifications in generalized topology, Topology Appl. 194, 241–268, 2015.
  • [21] A. Piękosz, and E. Wajch, Quasi-Metrizability of Bornological Biuniverses in ZF, J. Convex Anal. 22 (4), 1041–1060, 2015.
  • [22] S. Salbany, Bitopological spaces, compactifications and completions, in: Math. Monographs, Univ. of Cape Town, Cape Town, 1974.
  • [23] T. Vroegrijk, Pointwise bornological spaces, Topology Appl. 156, 2019–2027, 2009.
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0002-7515-2418
Author: Artur PİĘKOSZ (Primary Author)
Institution: Cracow University of Technology
Country: Poland


Orcid: 0000-0003-1864-2303
Author: Eliza WAJCH
Institution: Siedlce University of Natural Sciences and Humanities
Country: Poland


Dates

Publication Date : December 8, 2019

Bibtex @research article { hujms656663, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2019}, volume = {48}, pages = {1653 - 1666}, doi = {}, title = {Bornological quasi-metrizability in generalized topology}, key = {cite}, author = {PİĘKOSZ, Artur and WAJCH, Eliza} }
APA PİĘKOSZ, A , WAJCH, E . (2019). Bornological quasi-metrizability in generalized topology. Hacettepe Journal of Mathematics and Statistics , 48 (6) , 1653-1666 . Retrieved from https://dergipark.org.tr/en/pub/hujms/issue/50516/656663
MLA PİĘKOSZ, A , WAJCH, E . "Bornological quasi-metrizability in generalized topology". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1653-1666 <https://dergipark.org.tr/en/pub/hujms/issue/50516/656663>
Chicago PİĘKOSZ, A , WAJCH, E . "Bornological quasi-metrizability in generalized topology". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1653-1666
RIS TY - JOUR T1 - Bornological quasi-metrizability in generalized topology AU - Artur PİĘKOSZ , Eliza WAJCH Y1 - 2019 PY - 2019 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1653 EP - 1666 VL - 48 IS - 6 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2018 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Bornological quasi-metrizability in generalized topology %A Artur PİĘKOSZ , Eliza WAJCH %T Bornological quasi-metrizability in generalized topology %D 2019 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 48 %N 6 %R %U
ISNAD PİĘKOSZ, Artur , WAJCH, Eliza . "Bornological quasi-metrizability in generalized topology". Hacettepe Journal of Mathematics and Statistics 48 / 6 (December 2019): 1653-1666 .
AMA PİĘKOSZ A , WAJCH E . Bornological quasi-metrizability in generalized topology. Hacettepe Journal of Mathematics and Statistics. 2019; 48(6): 1653-1666.
Vancouver PİĘKOSZ A , WAJCH E . Bornological quasi-metrizability in generalized topology. Hacettepe Journal of Mathematics and Statistics. 2019; 48(6): 1666-1653.