Topology may be considered as an abstract study of the limit point concept. As such, it stems in part from recognition of the fact that many important mathematical topics depend entirely upon the properties of limit points. This study shows that compactness implies limit point compactness but not conversely and every compact space is locally compact but not conversely. This study also shows that compactness, limit point compactness and sequentially compactness are equivalent in metrizable spaces and the product of finitely many compact spaces is a locally compact space. This study introduce it here as an interesting application of the Tychonoff theorem.
| Published in | Pure and Applied Mathematics Journal (Volume 3, Issue 5) |
| DOI | 10.11648/j.pamj.20140305.13 |
| Page(s) | 105-112 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2014. Published by Science Publishing Group |
Metric Spaces, Topological Space, Compact Space, Locally Compact Space, Sequentially Compactness, Neighborhood
| [1] | Seymour Lipschutz, General Topology, McGraw-Hill Book Company, Singapore, 1965. |
| [2] | Munkres, James R, Topology, 2nd edition, Prentice Hall, 2000. |
| [3] | John L. Kelley, General topology, Van Nostrand, 1955. |
| [4] | George F. Simmons, Topology and Modern Analysis, McGraw-Hill, Inc. 1963. |
| [5] | Mitra, M., Study of some properties of topological spaces and of their generalizations, Ph.D. Thesis, Rajshahi University, 2006. |
| [6] | Bert Mendelson, Introduction to Topology, Allynand and Bacon, Inc. U.S.A., 1985. |
| [7] | N.D. Gautam and Shanti Narayan, General Topology, 1976. |
| [8] | K.D. Joshi, Introduction to General Topology, 1983 |
| [9] | J V Deshpabnde, Introductory to Topology,Centre of advanced study in Mathematics, University of Bombay. |
| [10] | John G Hocking and Gall S Young, Topology, Addison Weslesy, New York, 1961 |
| [11] | Steven A. Gall, Point Set Topology, 1964. |
| [12] | Dugundji, J., Topology, Wm. C. Brown Publisher, 1989 (Reprint New Delhi, 1995). |
APA Style
Rabeya Akter, Nour Mohammed Chowdhury, Mohammad Safi Ullah. (2014). A Study on Compactness in Metric Spaces and Topological Spaces. Pure and Applied Mathematics Journal, 3(5), 105-112. https://doi.org/10.11648/j.pamj.20140305.13
ACS Style
Rabeya Akter; Nour Mohammed Chowdhury; Mohammad Safi Ullah. A Study on Compactness in Metric Spaces and Topological Spaces. Pure Appl. Math. J. 2014, 3(5), 105-112. doi: 10.11648/j.pamj.20140305.13
AMA Style
Rabeya Akter, Nour Mohammed Chowdhury, Mohammad Safi Ullah. A Study on Compactness in Metric Spaces and Topological Spaces. Pure Appl Math J. 2014;3(5):105-112. doi: 10.11648/j.pamj.20140305.13
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.pamj.20140305.13,
author = {Rabeya Akter and Nour Mohammed Chowdhury and Mohammad Safi Ullah},
title = {A Study on Compactness in Metric Spaces and Topological Spaces},
journal = {Pure and Applied Mathematics Journal},
volume = {3},
number = {5},
pages = {105-112},
doi = {10.11648/j.pamj.20140305.13},
url = {https://doi.org/10.11648/j.pamj.20140305.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140305.13},
abstract = {Topology may be considered as an abstract study of the limit point concept. As such, it stems in part from recognition of the fact that many important mathematical topics depend entirely upon the properties of limit points. This study shows that compactness implies limit point compactness but not conversely and every compact space is locally compact but not conversely. This study also shows that compactness, limit point compactness and sequentially compactness are equivalent in metrizable spaces and the product of finitely many compact spaces is a locally compact space. This study introduce it here as an interesting application of the Tychonoff theorem.},
year = {2014}
}
TY - JOUR T1 - A Study on Compactness in Metric Spaces and Topological Spaces AU - Rabeya Akter AU - Nour Mohammed Chowdhury AU - Mohammad Safi Ullah Y1 - 2014/10/20 PY - 2014 N1 - https://doi.org/10.11648/j.pamj.20140305.13 DO - 10.11648/j.pamj.20140305.13 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 105 EP - 112 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20140305.13 AB - Topology may be considered as an abstract study of the limit point concept. As such, it stems in part from recognition of the fact that many important mathematical topics depend entirely upon the properties of limit points. This study shows that compactness implies limit point compactness but not conversely and every compact space is locally compact but not conversely. This study also shows that compactness, limit point compactness and sequentially compactness are equivalent in metrizable spaces and the product of finitely many compact spaces is a locally compact space. This study introduce it here as an interesting application of the Tychonoff theorem. VL - 3 IS - 5 ER -
Information
Email: