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 and studied. 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.