Volume 2, Issue 2, June 2017, Page: 54-58
On Analytical Approach to Semi-Open/Semi-Closed Sets
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
Received: Jan. 26, 2017;       Accepted: Feb. 16, 2017;       Published: Mar. 3, 2017
DOI: 10.11648/j.dmath.20170202.15      View  1570      Downloads  74
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.
Keywords
Open and Closed Sets, Semi-Open /Semi-Closed Sets, Metric Spaces, Topological Spaces
To cite this article
Musundi Sammy Wabomba, Kinyili Musyoka, Priscah Moraa Ohuru, On Analytical Approach to Semi-Open/Semi-Closed Sets, International Journal of Discrete Mathematics. Vol. 2, No. 2, 2017, pp. 54-58. doi: 10.11648/j.dmath.20170202.15
Copyright
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[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.
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