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Keywords:
zero-dimensional space; strongly zero-dimensional space; $\mathbb{N}$-compact space; Banaschewski compactification; pseudocompact space; functionally countable subalgebra; support; cellularity; remainder; almost $P$-space; Parovičenko space
Summary:
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., $C^{\psi}_c(X)=M_c^{\beta_0X\setminus \upsilon_0X}$), where $\beta_0X$ is the Banaschewski compactification of $X$ and $\upsilon_0X$ is the $\mathbb{N}$-compactification of $X$. This implies that for an $\mathbb{N}$-compact space $X$, the intersection of all free maximal ideals in $C_c(X)$ is equal to $C_c^K(X)$, i.e., $M_c^{\beta_0X\setminus X}=C_c^K(X)$. By applying methods of functionally countable subalgebras, we then obtain some results in the remainder of the Banaschewski compactification. We show that for a non-pseudocompact zero-dimensional space $X$, the set $\beta_0X\setminus \upsilon_0X$ has cardinality at least $2^{2^{\aleph_0}}$. Moreover, for a locally compact and $\mathbb{N}$-compact space $X$, the remainder $\beta_0X\setminus X$ is an almost $P$-space. These results lead us to find a class of Parovičenko spaces in the Banaschewski compactification of a non pseudocompact zero-dimensional space. We conclude with a theorem which gives a lower bound for the cellularity of the subspaces $\beta_0X\setminus \upsilon_0X$ and $\beta_0X\setminus X$, whenever $X$ is a zero-dimensional, locally compact space which is not pseudocompact.
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