Document Type : Original Manuscript
Faculty of Petroleum and Gas, Yasouj University, Gachsaran, Iran.
In this paper, we consider the $m_c$-topology on $C_c(X)$, the functionally countable subalgebra of $C(X)$. We show that a Tychonoff space $X$ is countably pseudocompact if and only if the $m_c$-topology and the $u_c$-topology on $C_c(X)$ coincide. It is shown that whenever $X$ is a zero-dimensional space, then $C_c(X)$ is first countable if and only if $C(X)$ with the $m$-topology is first countable. Also, the set of all zero-divisors of $C_c(X)$ is closed if and only if $X$ is an almost $P$-space. We show that if $X$ is a strongly zero-dimensional space and $U$ is the set of all units of $C_c(X)$, then the maximal ring of quotients of $C_c(U)$ and $C_c(C_c(X))$ are isomorphic.
3. R. Engelking, General Topology, Heldermann Verlag Berlin, (1989).
4. A. A. Estaji, A. Karimi Feizabadi and M. Zarghani, Zero elements and z-ideals in modified pointfree topology, Bull. Iran. Math. Soc., 43(7) (2017), 2205–2226.
5. N. J. Fine, L. Gillman and J. Lambek, Rings of quotients of rings of functions, Lecture Notes Series Mc-Gill University Press, Montreal, (1965).
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. —————–, C(X) versus its functionally countable subalgebra, Bull. Iran. Math. Soc., 45(1) (2019), 173–187.
8. L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, (1976).
9. E. Hewitt, Rings of real-valued continuous functions, I, Trans. Amer. Math. Soc., 64 (1948), 45–99.
10. O. A. S. Karamzadeh and Z. Keshtkar, On c-realcompact spaces, Quaest. Math., 42(8) (2018), 1135–1167.
11. O. A. S. Karamzadeh, M. Namdari and S. Soltanpour, On the locally functionally countable subalgebra of C(X), Appl. Gen. Topol., 16(2) (2015), 183–207.
12. G. D. Maio, L. Hola, D. Holy and R. A. McCoy, Topologies on the space of continuous functions, Topology Appl., 86 (1998), 105–122.
13. M. A. Mulero, Algebraic properties of rings of continuous functions, Fund. Math., 149 (1996), 55–66.
14. M. Namdari and A. Veisi, Rings of quotients of the subalgebra of C(X) consisting of functions with countable image, Int. Math. Forum, 7 (2012), 561–571.
15. A. Veisi, The subalgebras of the functionally countable subalgebra of C(X), Far East J. Math. Sci. (FJMS), 101(10) (2017), 2285–2297.
16. A. Veisi, Invariant norms on the subalgebras of C(X) consisting of bounded
functions with countable image, JP Journal of Geometry and Topology, 21(3) (2018), 167–179.
17. A. Veisi, ec-Filters and ec-ideals in the functionally countable subalgebra of C(X), Appl. Gen. Topol., 20(2) (2019), 395–405.