%0 Journal Article
%T A NOTE ON Cc(X) VIA A TOPOLOGICAL RING
%J Journal of Algebraic Systems
%I Shahrood University of Technology
%Z 2345-5128
%A Mohamadian, R.
%A Namdari, M.
%A Najafian, H.
%A Soltanpour, S.
%D 2023
%\ 01/01/2023
%V 10
%N 2
%P 323-334
%! A NOTE ON Cc(X) VIA A TOPOLOGICAL RING
%K Functionally countable subalgebra
%K $m_c$-topology
%K $u_c$-topology
%K $N$-space
%R 10.22044/jas.2022.11467.1579
%X Let $C_c(X)$ (resp., $C_c^*(X)$) denote the functionallycountable 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.
%U https://jas.shahroodut.ac.ir/article_2477_e924e7f0f47be03484e4067a481fe8a8.pdf