A Wallman-Shanin-Type Compactification for Approach Spaces

Löwen, R.; Sioen, Mark

doi:10.1216/rmjm/1021477356

Cited by 2 publications

(14 citation statements)

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“…Working in the setting of approach spaces remedies this fact. It is known that in this broader setting both a Čech-Stone compactification β * (X) and a Wallman compactification W * (X) can be constructed in such a way that their approach distance induces the original approach distance of the given metric on X [23], [24]. Section 4 and the following ones, are a contribution to the compactification theory for approach spaces.…”

Section: Introductionmentioning

confidence: 99%

“…In [24] the Wallman compactification was introduced for the subcategory of App, consisting of all weakly symmetric T 1 -spaces. This class contains all T 2 uniform approach spaces.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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confidence: 99%

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confidence: 99%

“…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.…”

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confidence: 99%

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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%

“…In [24] the Wallman compactification was introduced for the subcategory of App, consisting of all weakly symmetric T 1 -spaces. This class contains all T 2 uniform approach spaces.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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