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

Let X be a zero-dimensional space and Cc(X) denote the functionally countable subalgebra of C(X). It is well known that β0X (the Banaschewski compactfication of X) is a quotient space of βX. In this article, we investigate a construction of β0X via βX by using Cc(X) which determines the quotient space of βX homeomorphic to β0X. Moreover, the construction of v 0X via vCcX (the subspace {p 2 βX : 8f 2 Cc(X), f *(p) ∞} of βX) is also investigated.