An Isomorphism of the Wallman and Čech-Stone compactifications

For a metrizable topological space X it is well known that in general the Čech-Stone compactification β(X) or Wallman W(X) are not metrizable. To remedy this fact one can alternatively associate point-set distance to metric, so called approach distance. It setting both β*(X) and W*(X) be constructed such way their distances induce original of metric on [23], [24].The main goal paper formulate necessary sufficient conditions for an isomorphic, thus answering question first raised [24]. The clue reach settle left open [11], compact normal. In particular result shows uniform T2 space, always We prove normal if only normal, we produce example showing that, unlike spaces, normality isomorphic. introduce strengthening regularity condition X, which call ideal-regularity, our theorem conclude ideal-regular, T1 with Classical results recovered implications (quasi-)metric spaces investigated.