Abstract

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 [11], 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 T₂ 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 T₁ if and only if X is a uniform T₁ approach space with β*(X) and W*(X) isomorphic. Classical topological results are recovered and implications for (quasi-)metric spaces are investigated.

摘要

对于可度量的拓扑空间X，众所周知，通常 Čech-Stone 紧化β ( X ) 或 Wallman 紧化W ( X ) 是不可度量的。为了弥补这一事实，人们可以选择将点集距离与度量相关联，即所谓的接近距离。众所周知，在这种情况下，Čech-Stone 紧化β *( X ) 和 Wallman 紧化W *( X ) 都可以以这样的方式构造，即它们的接近距离会导致X上度量的原始接近距离[23 ]，[24]。

本文的主要目标是为逼近空间X制定必要和充分条件，使得 Čech-Stone 紧化β *( X ) 和 Wallman 紧化W *( X ) 是同构的，因此回答了在 [ 24]。达到这个目标的第一个线索是解决 [11] 中未解决的问题，为紧凑的逼近空间正常制定充分的条件。特别是结果表明，均匀T _{2空间的 Čech-Stone 紧化}β *( X )总是正常的。我们证明了 Wallman 紧化W *( X) 是正规的当且仅当X是正规的，并且我们产生一个例子表明，与拓扑空间不同，在方法中，设置X的正规性不足以使β *( X ) 和W *( X ) 是同构的。我们在X上引入了一个强化的正则性条件，我们称之为理想正则性，并且在我们的主要定理中，我们得出结论，当且仅当X是具有 β的均匀T _{1逼近空间时，} X是理想正则、正态和T ₁ *( X ) 和W *( X） 同构。恢复了经典拓扑结果并研究了（准）度量空间的含义。