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A Study on Compactness in Metric Spaces and Topological Spaces

Received: 16 September 2014     Accepted: 27 September 2014     Published: 20 October 2014
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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.

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

Keywords

Metric Spaces, Topological Space, Compact Space, Locally Compact Space, Sequentially Compactness, Neighborhood

References
[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).
Cite This Article
  • 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

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    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

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    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

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  • @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}
    }
    

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    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.
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Author Information
  • Department of Mathematics, Jagannath University, Dhaka, Bangladesh

  • Department of Mathematics, World University of Bangladesh, Dhaka, Bangladesh

  • Department of Mathematics, Comilla University, Comilla, Bangladesh

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