A Study on Compactness in Metric Spaces and Topological Spaces
Pure and Applied Mathematics Journal
Volume 3, Issue 5, October 2014, Pages: 105-112
Received: Sep. 16, 2014; Accepted: Sep. 27, 2014; Published: Oct. 20, 2014
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Authors
Rabeya Akter, Department of Mathematics, Jagannath University, Dhaka, Bangladesh
Nour Mohammed Chowdhury, Department of Mathematics, World University of Bangladesh, Dhaka, Bangladesh
Mohammad Safi Ullah, Department of Mathematics, Comilla University, Comilla, Bangladesh
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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.
Keywords
Metric Spaces, Topological Space, Compact Space, Locally Compact Space, Sequentially Compactness, Neighborhood
To cite this article
Rabeya Akter, Nour Mohammed Chowdhury, Mohammad Safi Ullah, A Study on Compactness in Metric Spaces and Topological Spaces, Pure and Applied Mathematics Journal. Vol. 3, No. 5, 2014, pp. 105-112. doi: 10.11648/j.pamj.20140305.13
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).
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