Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University 10.15672/HJMS.2018.636 Mathematical Sciences Matematik On bornology of extended quasi-metric spaces https://orcid.org/0000-0001-9593-7899 Otafudu Olivier Olela University of the Witwatersrand https://orcid.org/0000-0003-1802-8872 Toko Wilson B. University of the Witwatersrand https://orcid.org/0000-0002-1453-0403 Mukonda Danny Rusangu University 12 08 2019 48 6 1767 1777 01 29 2018 07 09 2018 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

Beer studied the structure of sets equipped with the extended metrics with a focus on bornologies. In the paper [A. Piekosz and E. Wajch, Quazi-metrizability of bornological biuniverses inZF, J. Convex Anal. 2015], Piekosz and Wajch extended the well-known Hu's Theorem on boundedness in a topological space (see [S.-T. Hu, Boundedness in a topological space, J. Math. Pures Appl. 1949]) to the framework of quasi-metric spaces. In this note, we continue the work of Piekosz and Wajch. We show that many results on bornology of extended metric spaces due to Beer do not use the symmetry axiom of the extended metric, with appropriate modifications they still hold in the context of extended $T_0$-quasi-metric spaces.

 Extended quasi-metric metric bornology universal space [1] G. Beer, The structure of extended Real-valued metric spaces, Set-Valued Var. Anal 21 (4), 591–602, 2013. [2] P. Fletcherand and W.F. Lindgren, Quasi-uniform spaces (Dekker, 1982). [3] S.T. Hu, Boundedness in a topological space, J. Math. Pures Appl. 228, 287–320, 1949. [4] K. Kemajou, H.P. Künzi and O. Olela Otafudu, The Isbell-hull of a di-space, Topol Appl. 159 (9), 2463–2475, 2012. [5] H.P. Künzi, An introduction to quasi-uniform spaces, Contemp. Math. 486, 239–304, 2009. [6] H.P. Künzi and C. Makitu Kivuvu, A double completion for an arbitrary T0-quasimetric space, J. Log. Algebr. Program. 76 (2), 251–269, 2008. [7] H.P. Künzi and O. Olela Otafudu, q-hyperconvexity in quasipseudometric spaces and fixed point theorems, J. Funct. Spaces Appl. Art.ID 765903, 18 pp., 2012, [8] M.G. Murdeshwar and K.K. Theckedath, Boundedness in a quasi-uniform space, Canad. Math. Bull. 13, 367–370, 1970. [9] A. Piekosz and E. Wajch, Quasi-metrizability of bornological biuniverses in ZF, J. Convex Anal. 22 (4), 1041–1060, 2015.