On the Behavior of Connectedness Properties in Isotonic Spaces under Perfect Mappings
ABSTRACT
The topological study of connectedness is heavily geometric or visual. Connectedness and connectedness-like properties play an important role in most topological characterization theorems, as well as in the study of obstructions to the extension of functions. In this paper, the behaviour of these properties in the realm of closure spaces is investigated using the class of perfect mappings. A perfect mapping is a type of map under which the image generally inherits the properties of the mapped space. It turns out that the general behaviour of connectedness properties in topological spaces extends to the class of isotone space.
Cite this paper
S. Gathigi, M. Gichuki and K. Sogomo, "On the Behavior of Connectedness Properties in Isotonic Spaces under Perfect Mappings," Advances in Pure Mathematics, Vol. 3 No. 9, 2013, pp. 689-691. doi: 10.4236/apm.2013.39093.
S. Gathigi, M. Gichuki and K. Sogomo, "On the Behavior of Connectedness Properties in Isotonic Spaces under Perfect Mappings," Advances in Pure Mathematics, Vol. 3 No. 9, 2013, pp. 689-691. doi: 10.4236/apm.2013.39093.
References
[1] J. L. Kelley, “General Topology,” Springer, New York, 1955. [2] A. S. Mashhour and M. H. Ghanim, “On Closure Spaces,” Indian Journal of Pure and Applied Mathematics, Vol. 106, No. 3, 1985, pp. 680-691. [3] W. J. Thron, “What Results are Valid on Cech-closure Spaces”, Topology Proceedings. Vol 6, 1981, pp. 135-158. [4] A. K. Elzenati and E. D. Habil, “Connectedness in Isotonic Spaces,” Turkish Journal of Mathematics, Vol. 30, No. 6 , 2006, pp. 247-262. [5] K. Kuratowski, “Introduction to Set Theory and Topology,” Pergamon Press Ltd., London, 1961. [6] B. Dai, “Strongly Connected Spaces,” National University of Singapore Journal, Vol. 12, No. 6, 2000, pp. 14-20. [7] E. Michael, “Bi-Quotient Maps and Cartesian Product of Quotient Maps,” Anneles De l’institut Fourier, Vol. 18, No. 2, 1968, pp. 287-302. [8] R. Engelking, “General Topology,” Haldermann Verlag, Berlin, 1989.
[1] J. L. Kelley, “General Topology,” Springer, New York, 1955. [2] A. S. Mashhour and M. H. Ghanim, “On Closure Spaces,” Indian Journal of Pure and Applied Mathematics, Vol. 106, No. 3, 1985, pp. 680-691. [3] W. J. Thron, “What Results are Valid on Cech-closure Spaces”, Topology Proceedings. Vol 6, 1981, pp. 135-158. [4] A. K. Elzenati and E. D. Habil, “Connectedness in Isotonic Spaces,” Turkish Journal of Mathematics, Vol. 30, No. 6 , 2006, pp. 247-262. [5] K. Kuratowski, “Introduction to Set Theory and Topology,” Pergamon Press Ltd., London, 1961. [6] B. Dai, “Strongly Connected Spaces,” National University of Singapore Journal, Vol. 12, No. 6, 2000, pp. 14-20. [7] E. Michael, “Bi-Quotient Maps and Cartesian Product of Quotient Maps,” Anneles De l’institut Fourier, Vol. 18, No. 2, 1968, pp. 287-302. [8] R. Engelking, “General Topology,” Haldermann Verlag, Berlin, 1989.