cc1(x) and related ideals in cc(x)

In this paper, we introduce and investigate the subring cc1(x), consisting of all continuous real-valued functions on x with countable image that vanish at infinity. this ring generalizes the classical c1(x) to the context of countable-valued functions. we prove that cc1(x) coincides with the intersection of all free ideals in c∗ c (x), the ring of bounded continuous countable-valued functions. moreover, we characterize when cc1(x) forms an ideal in cc(x), showing that this occurs if and only if x is c-pseudocompact. relationships between cc1(x) and other subrings, including cck(x) (functions with compact support) and cc (x) (functions with c-pseudocompact support), are studied, and we establish that cc1(x) = cc (x) if and only if x is compact. finally, we examine the ideal i(x), defined as the intersection of all free maximal ideals of cc(x), and describe various compactness properties|µ0-, η0-, and c -compactness|via inclusions among these ideals.