Classification of Weakly Infinite-Dimensional Spaces and Essential Mappings

摘要

A classification of weakly infinite-dimensional spaces is given by introducing the weak large transfinite dimension w-Ind. This function coincides with the covering dimension dim in the finite case. The classification provided by w-Ind is the same as the one given by Pol's (1983) index for weakly infinite-dimensional compact metric spaces. Results concerning the relation between w-Ind and essential mappings into Henderson's (1968) cubes J sup alpha, where alpha is a countable ordinal number are obtained. It is proved that for a compact metric space X, the following statements are equivalent: index X is greater than or = omega 0 sup alpha (ordinal exponentiation); w-Ind X is greater than or = alpha; there exists an essential mapping of X x C, the product of X with the Cantor set, to J sup alpha.