a note on cc(x) via a topological ring

Let cc(x) (resp., c ∗ c (x)) denote the functionally countable subalgebra of c(x) (resp., c∗ (x)), consisting of all functions (resp., bounded functions) with countable image. cc(x) (resp., c∗ c (x)) as a topological ring via mc-topology (resp.,m∗ c -topology) and uc-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 andstudied. it is shown that the mc-topology on cc(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.