An isomorphism of the Wallman and Čech-Stone compactifications

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 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.