### Mostafa Ghadermazi

University of Kurdistan, Sanandaj, Iran### O.A.S. Karamzadeh

Chamran University, Ahvaz, Iran### M. Namdari

Chamran University, Ahvaz, Iran

## 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.

## Cite this article

Mostafa Ghadermazi, O.A.S. Karamzadeh, M. Namdari, On the Functionally Countable Subalgebra of $C(X)$. Rend. Sem. Mat. Univ. Padova 129 (2013), pp. 47–69

DOI 10.4171/RSMUP/129-4