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

Mehdi Parsinia

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 ${pin beta X: forall fin C_c(X), f^*(p)<infty}$ of $beta X$) is also investigated.

Highlights

Throughout this article all topological spaces are assumed to be zero-dimensional
We investigate the construction of a topological space X for which βcX = βX from which it follows that τc may be properly contained in τ
We investigate the construction of the smallest c-realcompact space in which

Summary

Introduction

Throughout this article all topological spaces are assumed to be zero-dimensional (that is, are Hausdorff and contain a base of clopen sets). {1, 2} is a compact space, f has a continuous extension F : βX −→ {0, 1} It follows that Z(F ) ∩ X = Z(f ) and, as Z(F ) = F −1(0) is a clopen set in βX, we have clβX Z(f ) = Z(F ). ˆi is a continuous bijection from the compact βX ∼c to the Hausdorff space implies thati is a homeomorphism It follows from Lemma 3.12 that maximal ideals of Cc(X) are precisely the ideals {f ∈ Cc(X) : [p] ⊆ clβX Z(f )}, for p ∈ βX, which we denote by Mc[p] for each p ∈ βX. Let ξ be the homeomorphism from βcX ∼c onto whose existence just proved in the proof of Theorem 3.4.

It thus follows that λ

Then the following