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The Study of the Concept of Q*Compact Spaces

Received: 24 October 2017     Accepted: 15 November 2017     Published: 2 February 2018
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Abstract

The aim of this research is to extend the new type of compact spaces called Q* compact spaces, study its properties and generate new results of the space. It investigate the Q*-compactness of topological spaces with separable, Q*-metrizable, Q*-Hausdorff, homeomorphic, connected and finite intersection properties. The closed interval [0, 1] is Q* compact. So, it is deduced that the closed interval [0, 1] is Q*-compact. For example, if (X, τ) = ℝ and A = (0, ∞) then A is not Q*-compact. A subset S of ℝ is Q*-compact. Also, if (X, τ) is a Q*-compact metrizable space. Then (X, τ) is separable. (Y, τ1) is Q*-compact and metrizable if f is a continuous mapping of a Q*-compact metric space (X, d) onto a Q*-Hausdorff space (Y, τ1). An infinite subset of a Q*-compact space must have a limit point. The continuous mapping of a Q*-compact space has a greatest element and a least element. Eleven theorems were considered and their results were presented accordingly.

Published in Pure and Applied Mathematics Journal (Volume 7, Issue 1)
DOI 10.11648/j.pamj.20180701.11
Page(s) 1-5
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), 2018. Published by Science Publishing Group

Keywords

Topological Paces, Semi Compact Spaces, Q*O Compact Space

References
[1] P. Padma, S. Udayakumar, (January 2015) On Q*O Compact spaces, Journal of Progressive Research in Mathematics (JPRM) ISSN: 2395-0218 Volume 1, Issue 1.
[2] P. Padma, S. Udayakumar, (January 2015) On Q*s - regular spaces, Journal of Progressive Research in Mathematics (JPRM) ISSN: 2395-0218 Volume 1, Issue 1.
[3] P. Padma, K. Chandrasekhararao and S. Udayakumar, (2013) “Pairwise SC compact spaces”, International Journal of Analysis and Applications, Volume 2, Number 2, 162-172.
[4] M. Murugalingam and N. Laliltha, ( 2010) “Q star sets”, Bulletin of pure and applied Sciences, Volume 29E Issue 2 p. 369-376.
[5] M. Murugalingam and N. Laliltha, (2011) “Properties of Q* sets”, Bulletin of pure and applied Sciences, Volume 3E Issue 2 p. 267-277.
[6] P. Padma and S. Udayakumar, (2011) “Pairwise Q* separation axioms in bitopological spaces” International Journal of Mathematical Achieve – 3 (12), 2012, 4959-4971.
[7] P. Padma and S. Udayakumar, (Jul - Aug 2012) “τ1τ2- Q* continuous maps in bitopological spaces” Asian Journal of Current Engineering and Mathematics, 1:4, 227-229.
[8] Sidney A. Morris, (2011) “Topology without Tears”, version of January 2 2011. P. 155-167.
[9] Tom Benidec, Chris Best and Michael Bliss, (2007) “Introduction to Topology”, Renzo’s Math 470, Winter, p. 24-55.
[10] Kannan. K., (2009) “Contribution to the study of some generalized closed sets in bitopological spaces, (Ph.D Thesis).
[11] V. K. Sharma: (1990) A study of some separation and covering axioms in topological and bitopological spaces. Ph. D. Thesis, Meerut Univ.
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  • APA Style

    Ibrahim Bassi, Yakubu Gabriel, Onuk Oji Galadima. (2018). The Study of the Concept of Q*Compact Spaces. Pure and Applied Mathematics Journal, 7(1), 1-5. https://doi.org/10.11648/j.pamj.20180701.11

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    ACS Style

    Ibrahim Bassi; Yakubu Gabriel; Onuk Oji Galadima. The Study of the Concept of Q*Compact Spaces. Pure Appl. Math. J. 2018, 7(1), 1-5. doi: 10.11648/j.pamj.20180701.11

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    AMA Style

    Ibrahim Bassi, Yakubu Gabriel, Onuk Oji Galadima. The Study of the Concept of Q*Compact Spaces. Pure Appl Math J. 2018;7(1):1-5. doi: 10.11648/j.pamj.20180701.11

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  • @article{10.11648/j.pamj.20180701.11,
      author = {Ibrahim Bassi and Yakubu Gabriel and Onuk Oji Galadima},
      title = {The Study of the Concept of Q*Compact Spaces},
      journal = {Pure and Applied Mathematics Journal},
      volume = {7},
      number = {1},
      pages = {1-5},
      doi = {10.11648/j.pamj.20180701.11},
      url = {https://doi.org/10.11648/j.pamj.20180701.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20180701.11},
      abstract = {The aim of this research is to extend the new type of compact spaces called Q* compact spaces, study its properties and generate new results of the space. It investigate the Q*-compactness of topological spaces with separable, Q*-metrizable, Q*-Hausdorff, homeomorphic, connected and finite intersection properties. The closed interval [0, 1] is Q* compact. So, it is deduced that the closed interval [0, 1] is Q*-compact. For example, if (X, τ) = ℝ and A = (0, ∞) then A is not Q*-compact. A subset S of ℝ is Q*-compact. Also, if (X, τ) is a Q*-compact metrizable space. Then (X, τ) is separable. (Y, τ1) is Q*-compact and metrizable if f is a continuous mapping of a Q*-compact metric space (X, d) onto a Q*-Hausdorff space (Y, τ1). An infinite subset of a Q*-compact space must have a limit point. The continuous mapping of a Q*-compact space has a greatest element and a least element. Eleven theorems were considered and their results were presented accordingly.},
     year = {2018}
    }
    

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    AB  - The aim of this research is to extend the new type of compact spaces called Q* compact spaces, study its properties and generate new results of the space. It investigate the Q*-compactness of topological spaces with separable, Q*-metrizable, Q*-Hausdorff, homeomorphic, connected and finite intersection properties. The closed interval [0, 1] is Q* compact. So, it is deduced that the closed interval [0, 1] is Q*-compact. For example, if (X, τ) = ℝ and A = (0, ∞) then A is not Q*-compact. A subset S of ℝ is Q*-compact. Also, if (X, τ) is a Q*-compact metrizable space. Then (X, τ) is separable. (Y, τ1) is Q*-compact and metrizable if f is a continuous mapping of a Q*-compact metric space (X, d) onto a Q*-Hausdorff space (Y, τ1). An infinite subset of a Q*-compact space must have a limit point. The continuous mapping of a Q*-compact space has a greatest element and a least element. Eleven theorems were considered and their results were presented accordingly.
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Author Information
  • Department of Mathematics, Federal University, Lafia, Nigeria

  • Department of Mathematics, Shepherd’s International College, Akwanga, Nigeria

  • Department of Physics, Federal University, Gashua, Nigeria

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