Shahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101A NOTE ON Cc(X) VIA A TOPOLOGICAL RING323334247710.22044/jas.2022.11467.1579ENR. MohamadianDepartment of Mathematics, Shahid Chamran University of Ahvaz, P.O. Box
6135783151, Ahvaz, Iran.000000033350366XM. NamdariDepartment of Mathematics, Shahid Chamran University of Ahvaz, P.O. Box
6135783151, Ahvaz, Iran.0000-0003-0966-7234H. NajafianDepartment of Mathematics, Shahid Chamran University of Ahvaz, P.O. Box
6135783151, Ahvaz, Iran.S. SoltanpourDepartment of Science, Petroleum University of Technology, P.O. Box 6318714317,
Ahvaz, Iran.0000-0002-1072-9845Journal Article20211209Let $C_c(X)$ (resp., $C_c^*(X)$) denote the functionally<br />countable subalgebra of $C(X)$ (resp., $C^*(X)$),<br />consisting of all functions (resp., bounded functions) with countable image.<br />$C_c(X)$ (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. <br />Topological spaces which are called $N$-spaces are introduced and studied.<br />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.https://jas.shahroodut.ac.ir/article_2477_e924e7f0f47be03484e4067a481fe8a8.pdf