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International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 43 | Number 3 | Year 2017 | Article Id. IJMTT-V43P528 | DOI : https://doi.org/10.14445/22315373/IJMTT-V43P528

On Point Wise Products of Uniformly Continuous Functions on Uniform Space


Ku. S.B.Tadam, Dr. S.M.Padhye
Abstract

In this paper we obtain the sufficient conditions on a uniform space (x,u) for which uc(x), the family of all uniformly continuous functions on x is an algebra. It is proved that Theorem A: If (x,u) is a uniformly continuous uniform space then uc(x), is an algebra. Theorem B:If is precompact uniform space. Then is an algebra. We prove that ucR(x), the family of all uniformly continuous real valued functions on x is a lattice of functions which need not be a complete lattice in the sense that every subset of ucR(x) may not have supremum or infimum in ucR(x) by providing a counter example.

Keywords
Uniform space, Uniformly continuous space, Lattice, Complete Lattice.
References

1. J.L.Kelley, General Topology, Van Nostrand, Princeton, Toronto, Melbourne, London 1955.
2. Kundu, S. and Jain, T., Atsuji spaces: Equivalent conditions, Topology Proceedings, Vol.30, No.1, 2006, 301-325.
3. Russell C. Walker, The Stone-Cech Compactification, Springer-Verlag Berlin Heidelberg New York 1974
4. S.B. Nadler, Jr. and Donna M. Zitney, Point wise Products Of Uniformly Continuous Functions On Sets In The Real Line, The American Mathematical Monthly, Vol.114, No.2(Feb., 2007), pp.160-163.
5. W.J. Pervin, Foundations of General Topology, Academic Press Inc. New York, 1964.

Citation :

Ku. S.B.Tadam, Dr. S.M.Padhye, "On Point Wise Products of Uniformly Continuous Functions on Uniform Space," International Journal of Mathematics Trends and Technology (IJMTT), vol. 43, no. 3, pp. 234-237, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V43P528

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