Heredity of Lower Separation Axioms on Function Spaces
Author(s)
Njuguna E. Muturi*
ABSTRACT
The set of continuous functions from topological space Y to topological space Z endowed with a topology forms the function space. For A subset of Y, the set of continuous functions from the space A to the space Z forms the underlying function space with an induced topology. The function space has properties of topological space dependent on the properties of the space Z, such as the T0, T1, T2 and T3 separation axioms. In this paper, we show that the underlying function space inherits the T0, T1, T2 and T3 separation axioms from the function space, and that these separation axioms are hereditary on function spaces.
Cite this paper
Muturi, N. (2014) Heredity of Lower Separation Axioms on Function Spaces. Advances in Pure Mathematics, 4, 89-92. doi: 10.4236/apm.2014.43014.
Muturi, N. (2014) Heredity of Lower Separation Axioms on Function Spaces. Advances in Pure Mathematics, 4, 89-92. doi: 10.4236/apm.2014.43014.
References
[1] Fox, R.H. (1945) On Topologies for Function Spaces. American Mathematical Society, 27, 427-432. [2] Kelley, J.L. (1955) General Topology. Springer-Verlag, Berlin. [3] Muturi, N.E., Gichuki, M.N. and Sogomo, K.C. (2013) Topologies on the Underlying Function Space. International Journal of Management, IT and Engineering, 3, 101-113. [4] Arens, R.F. (1946) A Topology for Spaces of Transformations. The Annals of Mathematics, 2, 480-495.
http://dx.doi.org/10.2307/1969087 [5] Willard, S. (1970) General Topology. Addison-Wesley Publishing Company, United States of America. [6] Kathryn, F.P. (1993) The Open-Open Topology for Function Spaces. International Journal Mathematics and Mathematical Sciences, 16, 111-116. http://dx.doi.org/10.1155/S0161171293000134 [7] Muturi, N.E. (2014) Homeomorphism between the Underlying Function Space and the Subspace of the Function Space. Journal of Advanced Studies in Topology, 1, 57-60.
[1] Fox, R.H. (1945) On Topologies for Function Spaces. American Mathematical Society, 27, 427-432. [2] Kelley, J.L. (1955) General Topology. Springer-Verlag, Berlin. [3] Muturi, N.E., Gichuki, M.N. and Sogomo, K.C. (2013) Topologies on the Underlying Function Space. International Journal of Management, IT and Engineering, 3, 101-113. [4] Arens, R.F. (1946) A Topology for Spaces of Transformations. The Annals of Mathematics, 2, 480-495.
http://dx.doi.org/10.2307/1969087 [5] Willard, S. (1970) General Topology. Addison-Wesley Publishing Company, United States of America. [6] Kathryn, F.P. (1993) The Open-Open Topology for Function Spaces. International Journal Mathematics and Mathematical Sciences, 16, 111-116. http://dx.doi.org/10.1155/S0161171293000134 [7] Muturi, N.E. (2014) Homeomorphism between the Underlying Function Space and the Subspace of the Function Space. Journal of Advanced Studies in Topology, 1, 57-60.