ec-Filters and ec-ideals in the functionally countable subalgebra of C*(X)

Amir Veisi


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).


c-completely regular space; closed ideal; functionally countable space; ec-filter; ec-ideal; zero-dimensional space

Subject classification

54C30; 54C40; 54C05; 54G12; 13C11; 16H20.

Full Text:



F. Azarpanah, Intersection of essential ideals in C(X), Proc. Amer. Math. Soc. 125 (1997), 2149-2154.

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.

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.

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.

M. R. Koushesh, The Banach algebra of continuous bounded functions with separable support, Studia Mathematica 210, no. 3 (2012), 227-237.

R. Levy and M. D. Rice, Normal P-spaces and the $G_delta$-topology, Colloq. Math. 47 (1981), 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.

W. Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), 39-42.

A. Veisi, The subalgebras of the functionally countable subalgebra of C(X), Far East J. Math. Sci. (FJMS) 101, no. 10 (2017), 2285-2297.

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.

S. Willard, General Topology, Addison-Wesley, 1970.

Abstract Views

Metrics Loading ...

Metrics powered by PLOS ALM

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147