Research ArticleOpen AccessGoogle Scholar indexed
Some Classes of Bounded Sets in Quasi-Metric Spaces
School of Science and Technology Rusangu University Monze, Monze, Zambia
School of Science and Technology Rusangu University Monze, Monze, Zambia
Department of Mathematics and Statistics Mulungushi University, Kabwe, Zambia
- 1 School of Science and Technology Rusangu University Monze, Monze, Zambia
- 2 School of Science and Technology Rusangu University Monze, Monze, Zambia
- 3 Department of Mathematics and Statistics Mulungushi University, Kabwe, Zambia
Advances in Pure Mathematics·Volume 12 (2022)·Pages 701–714·Published 8 December 2022·DOI10.4236/apm.2022.1212053
Copy link · social · email
Abstract
This note deals with some classes of bounded subsets in a quasi-metric space. We study and compare the bounded sets, totally-bounded sets and the Bourbaki-bounded sets on quasi metric spaces. For example, we show that in a quasi-metric space, a set may be bounded but not totally bounded. In addition, we investigate their bornologies as well as their relationships with each other. For example, given a compatible quasi-metric, we intend to give some necessary and sufficient conditions for which a quasi metric bornology coincides with the bornology of totally bounded sets, the bornology of bourbaki bounded sets and bornology of bourbaki bounded subsets.
KeywordsQuasi-Metric-BoundednessTotally BoundednessBourbaki BoundednessBornology
- Otafudu, O.O. (2012) Convexity in Quasi-Metric Spaces. Ph.D. Thesis, University of Cape Town, Capetown.
- Cobzas, S. (2013) Functional Analysis in Asymmetric Normed Spaces. Springer Science & Business Media, Berlin. https://doi.org/10.1007/978-3-0348-0478-3
- Otafudu, O., Mukonda, D., et al. (2019) On Bornology of Extended Quasi-Metric Spaces. Hacettepe Journal of Mathematics and Statistics, 48, 1767-1777. https://doi.org/10.15672/HJMS.2018.636
- Garrido, M.I. and MeroNo, A.S. (2012) Some Classes of Bounded Sets in Metric Spaces. Contribucionesmatemáticasenhonor a Juan Tarrés. UCM, Madrid, pp. 179-186. https://eprints.ucm.es/id/eprint/17374/
- Mukonda, D. (2021) Bornological Aspects of Asymmetric Structure. Ph.D. Thesis, University of Witwatersrand, Johannesburg.
- Agyingi, C.A., Haihambo, P. and Künzi, H.-P.A. (2014) Endpoints in T0-Quasi-Metric Spaces. Topology and Its Applications, 168, 82-93. https://doi.org/10.1016/j.topol.2014.02.010
- Fletcher, P. and Lindgren, W. (1982) Quasi-Uniform Spaces, Lecture Notes in Pure and Applied Mathematics. New York.
- Mukonda, D. (2018) Bornology of Extended Quasi-Metric Spaces. MSc. Disertation, North-West University, Potchefstroom.
- Kubrusly, C.S. (2011) Elements of Operator Theory. Springer, Berlin. https://doi.org/10.1007/978-0-8176-4998-2
- Beer, G., Costantini, C. and Levi, S. (2011) Total Boundedness in Metrizable Spaces. Houston Journal of Mathematics, 37, 1347-1362.
- Minguzzi, E. (2012) Quasi-Pseudo-Metrization of Topological Preordered Spaces. Topology and Its Applications, 159, 2888-2898. https://doi.org/10.1016/j.topol.2012.05.029
- Beer, G. and Garrido, M.I. (2014) Bornologies and Locally Lipschitz Functions. Bulletin of the Australian Mathematical Society, 90, 257-263. https://doi.org/10.1017/S0004972714000215