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

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.

Keywords:

Functionally countable subalgebra , m_c-topology , pseudocompact space , zero-dimensional space