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On Analytical Approach to Semi-Open/Semi-Closed Sets

Received: 26 January 2017     Accepted: 16 February 2017     Published: 3 March 2017
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Abstract

The concept of open and closed sets has been extensively discussed on both metric and topological spaces. Various properties of these sets have been proved under the underlying spaces. However, scanty literature is available about semi-open /semi-closed sets on these spaces. For instance, little effort has been made in introducing these sets as clopen sets in topological spaces but no literature exists of the same under metric spaces. In this paper, with reference to the already existing definitions and properties of open and closed sets in metric spaces as well as in topological spaces we shall present definitions of semi-open/ semi-closed sets and furthermore prove basic properties of these sets on metrics spaces. The results of the study will provide a deeper understanding as well as extension knowledge for the concept of open and closed sets to their somewhat counter-intuitive terms of semi- open /semi-closed.

Published in International Journal of Discrete Mathematics (Volume 2, Issue 2)
DOI 10.11648/j.dmath.20170202.15
Page(s) 54-58
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), 2017. Published by Science Publishing Group

Keywords

Open and Closed Sets, Semi-Open /Semi-Closed Sets, Metric Spaces, Topological Spaces

References
[1] David R. Wilkins (2001). Course 212: Topological spaces. Hilary.
[2] Kenneth R. Davidson and Allan P. Donsig (2010). Undergraduate texts in: Real Analysis and Applications Mathematics. Theory in practice. Springer.
[3] Lecture 4 Econ 2001. (Aug13, 2015). Retrieved from http://www.pitt.edu/~luca/ECON2001/lecture_04.pdf.
[4] Metric spaces. (n.d). Retrieved from http://cseweb.ucsd.edu/~gill/CILASite/Resources/17AppABCbib.pdf.
[5] Metric Space Topology.(n.d).Retrieved from http://www.maths.uq.edu.au/courses/MATH3402/Lectures/topo.pdf.
[6] Murray H. Protter (1998). Basic Elements Real Analysis. Springer.
[7] Norman Levine, (Jan., 1963). Semi-Open Sets and Semi-Continuity in Topological Spaces. Mathematical Association of America, Vol. 70, No. 1, pp. 36-41.
[8] Open and closed sets. (n.d). Retrieved from http://www.u.arizona.edu/~mwalker/econ519/Econ519LectureNotes/OpenClosedSets.pdf .
[9] Royden, H. L., (1968). Real Analysis, 2nd edition, Macmillan, New York.
[10] Rudin, W., (1976). Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, New York.
Cite This Article
  • APA Style

    Musundi Sammy Wabomba, Kinyili Musyoka, Priscah Moraa Ohuru. (2017). On Analytical Approach to Semi-Open/Semi-Closed Sets. International Journal of Discrete Mathematics, 2(2), 54-58. https://doi.org/10.11648/j.dmath.20170202.15

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    ACS Style

    Musundi Sammy Wabomba; Kinyili Musyoka; Priscah Moraa Ohuru. On Analytical Approach to Semi-Open/Semi-Closed Sets. Int. J. Discrete Math. 2017, 2(2), 54-58. doi: 10.11648/j.dmath.20170202.15

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    AMA Style

    Musundi Sammy Wabomba, Kinyili Musyoka, Priscah Moraa Ohuru. On Analytical Approach to Semi-Open/Semi-Closed Sets. Int J Discrete Math. 2017;2(2):54-58. doi: 10.11648/j.dmath.20170202.15

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  • @article{10.11648/j.dmath.20170202.15,
      author = {Musundi Sammy Wabomba and Kinyili Musyoka and Priscah Moraa Ohuru},
      title = {On Analytical Approach to Semi-Open/Semi-Closed Sets},
      journal = {International Journal of Discrete Mathematics},
      volume = {2},
      number = {2},
      pages = {54-58},
      doi = {10.11648/j.dmath.20170202.15},
      url = {https://doi.org/10.11648/j.dmath.20170202.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170202.15},
      abstract = {The concept of open and closed sets has been extensively discussed on both metric and topological spaces. Various properties of these sets have been proved under the underlying spaces. However, scanty literature is available about semi-open /semi-closed sets on these spaces. For instance, little effort has been made in introducing these sets as clopen sets in topological spaces but no literature exists of the same under metric spaces. In this paper, with reference to the already existing definitions and properties of open and closed sets in metric spaces as well as in topological spaces we shall present definitions of semi-open/ semi-closed sets and furthermore prove basic properties of these sets on metrics spaces. The results of the study will provide a deeper understanding as well as extension knowledge for the concept of open and closed sets to their somewhat counter-intuitive terms of semi- open /semi-closed.},
     year = {2017}
    }
    

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  • TY  - JOUR
    T1  - On Analytical Approach to Semi-Open/Semi-Closed Sets
    AU  - Musundi Sammy Wabomba
    AU  - Kinyili Musyoka
    AU  - Priscah Moraa Ohuru
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    DO  - 10.11648/j.dmath.20170202.15
    T2  - International Journal of Discrete Mathematics
    JF  - International Journal of Discrete Mathematics
    JO  - International Journal of Discrete Mathematics
    SP  - 54
    EP  - 58
    PB  - Science Publishing Group
    SN  - 2578-9252
    UR  - https://doi.org/10.11648/j.dmath.20170202.15
    AB  - The concept of open and closed sets has been extensively discussed on both metric and topological spaces. Various properties of these sets have been proved under the underlying spaces. However, scanty literature is available about semi-open /semi-closed sets on these spaces. For instance, little effort has been made in introducing these sets as clopen sets in topological spaces but no literature exists of the same under metric spaces. In this paper, with reference to the already existing definitions and properties of open and closed sets in metric spaces as well as in topological spaces we shall present definitions of semi-open/ semi-closed sets and furthermore prove basic properties of these sets on metrics spaces. The results of the study will provide a deeper understanding as well as extension knowledge for the concept of open and closed sets to their somewhat counter-intuitive terms of semi- open /semi-closed.
    VL  - 2
    IS  - 2
    ER  - 

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Author Information
  • Department of Physical Sciences, Chuka University, Nairobi, Kenya

  • Department of Mathematics, Computer Science and Technology, University of Embu, Nairobi, Kenya

  • Department of Physical Sciences, Chuka University, Nairobi, Kenya

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