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

Document Type: Research Paper


Departemant of Mathematics, Shahid Chamran University of Ahvaz, Iran.


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.


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