On some non-Archimedean spaces of Alexandroff and Urysohn

Classical characterizations of four separable metrizable spaces are recalled, and generalized to classes of spaces which admit a uniformity with a totally ordered base. The Alexandroff-Urysohn characterization of the irrationals finds its closest analogues for strongly inaccessible cardinals, while the other three spaces, including the Cantor set, find their most natural analogues for weakly compact cardinals. In addition, A.H. Stoneʹs characterization of Baireʹs zero-dimensional spaces is extended to give internal characterizations of all spaces γλ × D, where D is discrete and γλ has the initial agreement topology. The historical background for the Alexandroff-Urysohn result is briefly surveyed.