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

### Mostafa Ghadermazi

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

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

Chamran University, Ahvaz, Iran

## Abstract

Let $C_{c}(X)={f∈C(X):f(X)is countable}$. Similar to $C(X)$ it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring $C_{c}(X)$ is either $C_{c}(X)$ or a semiprime (resp. prime) ideal in $C_{c}(X)$ . For an ideal $I$ in $C_{c}(X)$ , it is observed that $I$ and $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 $C_{c}(X)≅C_{c}(Y)$ . Consequently, if $X$ has only countable number of components, then $C_{c}(X)≅C(Y)$ for some zero-dimensional space $Y$. Spaces X for which $C_{c}(X)$ 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 $C_{c}(X)$ enjoys the algebraic properties of regularity, $ℵ_{_{0}}$-selfinjectivity and some others, whenever $C(X)$ has these properties. Finally an example of a space $X$ such that $C_{c}(X)$ 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