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

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 X-0 (the Banaschewski compactfication of X) is a quotient space of beta X. In this article, we investigate a construction of beta X-0 via beta X by using C-c(X) which determines the quotient space of beta X homeomorphic to beta X-0. Moreover, the construction of v(0)X via v(Cc) X (the subspace {p is an element of beta X : for all f is an element of C-c (X), f* (p) < infinity} of beta X) is also investigated.