Document Type : Original Manuscript


1 Department of Mathematics, Shahid Chamran University of Ahvaz, P.O. Box 6135783151, Ahvaz, Iran.

2 Department of Science, Petroleum University of Technology, P.O. Box 6318714317, Ahvaz, Iran.


Let $C_c(X)$ (resp., $C_c^*(X)$) denote the functionally
countable subalgebra of $C(X)$ (resp., $C^*(X)$),
consisting of all functions (resp., bounded functions) with countable image.
$C_c(X)$ (resp., $C_c^*(X)$) as a topological ring via $m_c$-topology (resp., $m^*_c$-topology) and $u_c$-topology (resp., $u^*_c$-topology) is investigated and the equality of the latter two topologies is characterized.
Topological spaces which are called $N$-spaces are introduced and studied.
It is shown that the $m_c$-topology on $C_c(X)$ and its relative topology as a subspace of $C(X)$ (with $m$-topology) coincide if and only if $X$ is an $N$-space. We also show that $X$ is pseudocompact if and only if it is both a countably pseudocompact, and an $N$-space.


1. S. K. Acharyya, R. Bharati, and A. D. Ray, Rings and subrings of continuous functions with countable range, Queast. Math., 44(6) (2021) 829–848.
2. F. Azarpanah, O. A. S. Karamzadeh, Z. Keshtkar, and A. R. Olfati, On maximal ideals of Cc(X) and the uniformity of its localization, Rocky Mt. J. Math., 48(2) (2018), 1–9.
3. F. Azarpanah, F. Manshoor, and R. Mohamadian, Connectedness and compactness in C(X) with the m-topology and generalized m-topology, Topology Appl., 159 (2012) 3486–3493.
4. E. van Douwen, Nonnormality or hereditary paracompactness of some spaces of real functions, Topology Appl., 39 (1991) 3–32.
5. R. Engelking, General Topology, Berlin, Germany, Heldermann Verlag, 1989.
6. M. Ghadermazi, O. A. S. Karamzadeh, and M. Namdari, On the functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova, 129 (2013), 47–69.
7. M. Ghadermazi, O. A. S. Karamzadeh, and M. Namdari, C(X) versus its functionally countable subalgebra, Bull. Iran. Math. Soc., 245 (2019), 173– 187.
8. L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, 1976.
9. J. Gomez-Pérez and W. W. McGovern, The m-topology on Cm(X) revisited, Topology Appl., 153 (2006) 1838–1848.
10. E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc., 48(64) (1948) 54–99.
11. G. Di Maio, L’. Holá, D. Holý, and D. McCoy, Topology on the space of continuous functions, Topology Appl., 86 (1998) 105–122.
12. A. Veisi, On the mc-topology on the functionally countable subalgebra of C(X), J. Algebr. Syst., 9(2) (2022) 335–345.