ec-Filters and ec-ideals in the functionally countable subalgebra of C*(X)
Keywords:c-completely regular space, closed ideal, functionally countable space, ec-filter, ec-ideal, zero-dimensional space
The purpose of this article is to study and investigate ec-filters on X and ec-ideals in C*c (X) in which they are in fact the counterparts of zc-filters on X and zc-ideals in Cc(X) respectively. We show that the maximal ideals of C*c (X) are in one-to-one correspondence with the ec-ultrafilters on X. In addition, the sets of ec-ultrafilters and zc-ultrafilters are in one-to-one correspondence. It is also shown that the sets of maximal ideals of Cc(X) and C*c (X) have the same cardinality. As another application of the new concepts, we characterized maximal ideals of C*c (X). Finally, we show that whether the space X is compact, a proper ideal I of Cc(X) is an ec-ideal if and only if it is a closed ideal in Cc(X) if and only if it is an intersection of maximal ideals of Cc(X).
F. Azarpanah, Intersection of essential ideals in C(X), Proc. Amer. Math. Soc. 125 (1997), 2149-2154. https://doi.org/10.1090/S0002-9939-97-04086-0
R. Engelking, General Topology, Heldermann Verlag Berlin, 1989.
A. A. Estaji, A. Karimi Feizabadi and M. Abedi, Zero-sets in point-free topology and strongly z-ideals, Bull. Iranian Math. Soc. 41, no. 5 (2015), 1071-1084.
N. J. Fine, L. Gillman and J. Lambek, Rings of quotients of rings of functions, Lecture Notes Series Mc-Gill University Press, Montreal, 1966.
M. Ghadermazi, O. A. S. Karamzadeh and M. Namdari, On functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova 129 (2013), 47-69. https://doi.org/10.4171/RSMUP/129-4
L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, 1976.
M. Henriksen, R. Raphael and R. G. Woods, $SP$-scattered spaces; a new generalization of scattered spaces, Comment. Math. Univ. Carolin. 48, no. 3 (2007), 487-505.
O. A. S. Karamzadeh, M. Namdari and S. Soltanpour, On the locally functionally countable subalgebra of C(X), Appl. Gen. Topol. 16, no. 2 (2015), 183-207. https://doi.org/10.4995/agt.2015.3445
O. A. S. Karamzadeh and M. Rostami, On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93 (1985), 179-184. https://doi.org/10.2307/2044578
M. R. Koushesh, The Banach algebra of continuous bounded functions with separable support, Studia Mathematica 210, no. 3 (2012), 227-237. https://doi.org/10.4064/sm210-3-3
R. Levy and M. D. Rice, Normal P-spaces and the $G_delta$-topology, Colloq. Math. 47 (1981), 227-240. https://doi.org/10.4064/cm-44-2-227-240
M. A. Mulero, Algebraic properties of rings of continuous functions, Fund. Math. 149 (1996), 55-66.
M. Namdari and A. Veisi, Rings of quotients of the subalgebra of C(X) consisting of functions with countable image, Inter. Math. Forum 7 (2012), 561-571.
D. Rudd, On two sum theorems for ideals of C(X), Michigan Math. J. 17 (1970), 139-141. https://doi.org/10.1307/mmj/1029000423
W. Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), 39-42. https://doi.org/10.1090/S0002-9939-1957-0085475-7
A. Veisi, The subalgebras of the functionally countable subalgebra of C(X), Far East J. Math. Sci. (FJMS) 101, no. 10 (2017), 2285-2297. https://doi.org/10.17654/MS101102285
A. Veisi, Invariant norms on the functionally countable subalgebra of C(X) consisting of bounded functions with countable image, JP Journal of Geometry and Topology 21, no. 3 (2018), 167-179. https://doi.org/10.17654/GT021030167
S. Willard, General Topology, Addison-Wesley, 1970.
How to Cite
This journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.