@article { author = {Parsinia, Mehdi}, title = {Constructing the Banaschewski compactification through the functionally countable subalgebra of $C(X)$}, journal = {Categories and General Algebraic Structures with Applications}, volume = {14}, number = {1}, pages = {167-180}, year = {2021}, publisher = {Shahid Beheshti University}, issn = {2345-5853}, eissn = {2345-5861}, doi = {10.29252/cgasa.14.1.167}, 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}, url = {https://cgasa.sbu.ac.ir/article_87513.html}, eprint = {https://cgasa.sbu.ac.ir/article_87513_b8f15e9052fb623c491eaaabd5b1e01e.pdf} }