The Banaschewski compactification of an approach space is of Wallman-Shanin-type

Colebunders, Eva; Sioen, Mark

doi:10.2989/16073606.2020.1753846

Cited by 1 publication

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“…Given a weakly symmetric T 1 approach space X, its bounded lower regular function frame L X , is a particular so called Wallman base. The general concept of a Wallman base was recalled in [12]. From the Wallman base L X , in [24] an extension of X is obtained on the set W * (X) of all maximal zero ideals Φ over L X .…”

Section: Introductionmentioning

confidence: 99%

An isomorphism of the Wallman and Čech-Stone compactifications

Colebunders

Löwen

Sioen

2021

Quaestiones Mathematicae

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For a metrizable topological space X it is well known that in general the Čech-Stone compactification β(X) or the Wallman compactification W (X) are not metrizable. To remedy this fact one can alternatively associate a point-set distance to the metric, a so called approach distance. It is known that in this setting both a Čech-Stone compactification β * (X) and a Wallman compactification W * (X) can be constructed in such a way that their approach distances induce the original approach distance of the metric on X [23], [24].The main goal in this paper is to formulate necessary and sufficient conditions for an approach space X such that the Čech-Stone compactification β * (X) and the Wallman compactification W * (X) are isomorphic, thus answering a question first raised in [24]. The first clue to reach this goal is to settle a question left open in [10], to formulate sufficient conditions for a compact approach space to be normal. In particular the result shows that the Čech-Stone compactification β * (X) of a uniform T 2 space, is always normal. We prove that the Wallman compactification W * (X) is normal if and only if X is normal, and we produce an example showing that, unlike for topological spaces, in the approach setting normality of X is not sufficient for β * (X) and W * (X) to be isomorphic. We introduce a strengthening of the regularity condition on X, which we call ideal-regularity, and in our main theorem we conclude that X is ideal-regular, normal and T 1 if and only if X is a uniform T 1 approach space with β * (X) and W * (X) isomorphic. Classical topological results are recovered and implications for (quasi-)metric spaces are investigated.

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Section: Introductionmentioning

confidence: 99%