Constructing the Banaschewski compactification through the functionally countable subalgebra of $C(X)$

Abstract

Let $X$ be a zero-dimensional space and $C_c(X)$ denote the functionally countable subalgebra of $C(X)$. It is well known that $\beta_0X$ (the Banaschewski compactfication of $X$) is a quotient space of $\beta X$. In this article, we investigate a construction of $\beta_0X$ via $\beta X$ by using $C_c(X)$ which determines the quotient space of $\beta X$ homeomorphic to $\beta_0X$. Moreover, the construction of $\upsilon_0X$ via $\upsilon_{_{C_c}}X$ (the subspace $\{p\in \beta X: \forall f\in C_c(X), f^*(p)<\infty\}$ of $\beta X$) is also investigated.

Keywords

• zero-dimensional space
• functionally countable subalgebra
• stone-$rm{check{c}}$ech compactification
• banaschewski compactification