A NOTE ON Cc(X) VIA A TOPOLOGICAL RING

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.

Keywords:

Functionally countable subalgebra , m_c-topology , u_c-topology , N-space