ON THE m_c-TOPOLOGY ON THE FUNCTIONALLY COUNTABLE SUBALGEBRA OF C(X)

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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.