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

In this paper, we consider the mc-topology on Cc(X), the functionally countable subalgebra of C(X). We show that a Tychonoff space X is countably pseudocompact if and only if the mc-topology and the uc-topology on Cc(X) coincide. It is shown that whenever X is a zero-dimensional space, then Cc(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 Cc(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 Cc(X), then the maximal ring of quotients of Cc(U) and Cc(Cc(X)) are isomorphic.