D-11 - Dyadic Compacta

Abstract

A compact Hausdorff topological space is called dyadic if it is a continuous image of a Cantor cube Dm, where m is some infinite cardinal number. This notion was inspired by the theorem of P.S. Alexandroff, announced in 1925, which states that every compact metric space is a continuous image of the Cantor set Dω0. Shanin introduced the notion that an uncountable cardinal number n is a caliber of a topological space for any family of cardinality n of nonempty open sets provided a subfamily of the same cardinality with nonempty intersection exists. A.S. Esenin–Volpin confirmed the natural conjecture posed by P.S. Alexandroff that a dyadic compactum satisfying the first axiom of countability is metrizable. He proved that the weight of a dyadic compactum is equal to the supremum of characters of all points of the original compactum. The Boolean algebra RO(X) of regular open sets of a dyadic space X is similar to the corresponding algebras of Cantor cubes. This chapter provides the background for different generalizations of the concept of dyadic spaces.