On the Functionally Countable Subalgebra of $C(X)$

Let Cc(X) = {f ∈ C(X) : f(X) is countable}. Similar to C(X) it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring Cc(X) is either Cc(X) or a semiprime (resp. prime) ideal in Cc(X). For an ideal I in Cc(X), it is observed that I and √ I have the same largest zc-ideal. If X is any topological space, we show that there is a zero-dimensional space Y such that Cc(X) ∼= Cc(Y ). Consequently, if X has only countable number of components, then Cc(X) ∼= C(Y ) for some zero-dimensional space Y . Spaces X for which Cc(X) is regular (called CP -spaces) are characterized both algebraically and topologically and it is shown that P -spaces and CP -spaces coincide when X is zero-dimensional. In contrast to C∗(X), we observe that Cc(X) enjoys the algebraic properties of regularity, א0selfinjectivity and some others, whenever C(X) has these properties. Finally an example of a space X such that Cc(X) is not isomorphic to any C(Y ) is given.