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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2019 / Volume: 48 Issue: 6

On bornology of extended quasi-metric spaces

Olivier Olela Otafudu(University of the Witwatersrand)0000-0001-9593-7899
Wilson B. Toko(University of the Witwatersrand)0000-0003-1802-8872
Danny Mukonda(Rusangu University)0000-0002-1453-0403
Sayfa
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Yayın Tarihi
2019-12-08
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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.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Beer studied the structure of sets equipped with the extended metrics with a focus on bornologies.
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Temel Bulgular & Veri:Convex Anal. 2015], Piekosz and Wajch extended the well-known Hu's Theorem on boundedness in a topological space (see [S.-T.
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Sonuç & Katkı: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.
🔬Tür:Research Article
📍Kapsam:2015 Verileri
⚙️Değişkenler:Extended quasi-metric · metric bornology · universal space

Abstract

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.

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Bu Makaleye Atıf Yap

Otafudu, O. O., Toko, W. B., Mukonda, D. (2019). On bornology of extended quasi-metric spaces. Hacettepe Journal of Mathematics and Statistics, Volume: 48 Issue: 6. https://doi.org/10.15672/HJMS.2018.636

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Kaynakça (9)

  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.