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

Abstract

Let {C_c{{\char 40}}X{{\char 41}}} =\{\,f\in C(X): f(X) \hbox{ is countable}\} . Similar to C(X) it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring {C_c{{\char 40}}X{{\char 41}}} is either {C_c{{\char 40}}X{{\char 41}}} or a semiprime (resp. prime) ideal in {C_c{{\char 40}}X{{\char 41}}} . For an ideal I in {C_c{{\char 40}}X{{\char 41}}} , it is observed that I and \sqrt{I} have the same largest z_c -ideal. If X is any topological space, we show that there is a zero-dimensional space Y such that {C_c{{\char 40}}X{{\char 41}}} \cong {C_c{{\char 40}}Y{{\char 41}}} . Consequently, if X has only countable number of components, then {C_c{{\char 40}}X{{\char 41}}} \cong C(Y) for some zero-dimensional space Y . Spaces X for which {C_c{{\char 40}}X{{\char 41}}} is regular (called CP -spaces) are characterized both algebraically and topo log ically and it is shown that P -spaces and CP -spaces coincide when X is zero-dimensional. In contrast to C^*(X) , we observe that {C_c{{\char 40}}X{{\char 41}}} enjoys the algebraic properties of regularity, \aleph _{_0} -selfinjectivity and some others, whenever C(X) has these properties. Finally an example of a space X such that {C_c{{\char 40}}X{{\char 41}}} is not isomorphic to any C(Y) is given.