Functionally countable subalgebras and some properties of Banaschewski compactification

Let $X$ be a zero-dimensional space and $C_c(X)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K(X)$ (resp., $C_{c}^{\psi}(X)$) the set of all functions in $C_c(X)$ with compact (resp., pseudocompact) support. First, we observe that $C_c^{K}(X)=O_c^{\beta_0X\setminus X}$ (resp., $C_c^{\psi}(X)=M_c^{\beta_0X\setminus \upsilon_0X}$). This implies that for an $\Bbb{N}$-compact space $X$, the intersection of all free maximal ideals in $C_c(X)$ equals to $C_c^K(X)$, i.e., $M_c^{\beta_0X\setminus X}=C_c^K(X)$. Afterwards, by applying methods of functionally countable subalgebras, we observe some results in the remainder of Banaschewski compactification. It is shown that for a zero-dimensional non pseudocompact space $X$, the set $\beta_0X\setminus \upsilon_0X$ has cardinality at least $2^{2^{\aleph_0}}$. Moreover, for a locally compact and $\Bbb{N}$-compact space $X$, the remainder $\beta_0X\setminus X$ is an almost $P$-space. These results leads us to find a class of Parovi$\breve{\mbox{c}}$enko spaces in Banaschewski compactification os a non pseudocompact zero-dimensional space. We conclude with a theorem which gives a lower bound for the cellularity of subspaces $\beta_0X\setminus \upsilon_0X$ and $\beta_0X\setminus X$, whenever $X$ is a zero-dimensional, locally compact space which is not pseudocompact.