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

Musundi Sammy Wabomba; Kinyili Musyoka; Priscah Moraa Ohuru

doi:doi:10.11648/j.dmath.20170202.15

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

Musundi Sammy Wabomba, Kinyili Musyoka, Priscah Moraa Ohuru

Published in International Journal of Discrete Mathematics (Volume 2, Issue 2)

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), 2024. 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, 2^nd edition, Macmillan, New York.
[10]	Rudin, W., (1976). Principles of Mathematical Analysis, 3^rd edition, McGraw-Hill, New York.

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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
Y1  - 2017/03/03
PY  - 2017
N1  - https://doi.org/10.11648/j.dmath.20170202.15
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

Musundi Sammy Wabomba

Department of Physical Sciences, Chuka University, Nairobi, Kenya
Kinyili Musyoka

Department of Mathematics, Computer Science and Technology, University of Embu, Nairobi, Kenya
Priscah Moraa Ohuru

Department of Physical Sciences, Chuka University, Nairobi, Kenya

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Plain Text BibTeX RIS

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - On Analytical Approach to Semi-Open/Semi-Closed Sets
AU  - Musundi Sammy Wabomba
AU  - Kinyili Musyoka
AU  - Priscah Moraa Ohuru
Y1  - 2017/03/03
PY  - 2017
N1  - https://doi.org/10.11648/j.dmath.20170202.15
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  -

Copy | Download