Abstract

For a topological zero-dimensional Hausdorff space (X, ) it is well known that the Banaschewski compactification ζ(X, ) is of Wallman-Shanin-type, meaning that there exists a closed basis (the collection of all clopen sets), such that the Wallman-Shanin compactification with respect to this closed basis is isomorphic to ζ(X, ).

For an approach space (X, ) the Wallman-Shanin compactification W (X, ) with respect to a Wallman-Shanin basis (a particular basis of the lower regular function frame ) was introduced by R. Lowen and the second author. Recently, various constructions of the Banaschewski compactification known for a topological space were generalised to the approach case. Given a Hausdorff zero-dimensional approach space (X, ), constructions of the Banaschewski compactification ζ^∗(X, ) were developed by the authors.

In this paper we construct a particular Wallman-Shanin basis for (X, ) and show that the Wallman-Shanin compactification with respect to this particular basis is isomorphic to ζ^∗(X, ).

摘要

对于拓扑零维的Hausdorff空间（X，）是公知的是，Banaschewski紧凑化ζ（X，）是沃尔曼-沙宁型的，这意味着存在一个封闭的基础（所有闭开集的集合），如关于这个封闭基的 Wallman-Shanin 紧化与ζ ( X , )同构。

对于逼近空间 ( X , ) ，R. Lowen 和第二作者介绍了关于 Wallman-Shanin 基（下正则函数框架的特定基）的 Wallman-Shanin 紧化W ( X , ) 。最近，以拓扑空间已知的 Banaschewski 紧化的各种构造被推广到方法案例。给定 Hausdorff 零维逼近空间 ( X , )，作者开发了 Banaschewski 紧化ζ ^∗ ( X , ) 的构造。

在本文中，我们为 ( X , )构造了一个特定的 Wallman-Shanin 基，并证明了关于这个特定基的 Wallman-Shanin 紧化与ζ ^∗ ( X , )同构。