Cohomological dimension of Tychonov spaces

The concept of the cohomological dimension dimG X is defined for any Tychonov (i.e., completely regular and Hausdorff) space X and any countable abelian group G. We prove that dimG X = dimG νX, where νX denotes the Hewitt realcompactification of X. We also give a spectral characterization of the dimension dimG. These two results allow us to reduce several problems concerning the dimension dimG of general spaces to the corresponding problems for Polish (i.e., completely metrizable and separable) spaces. For instance, we show that the inequality dimZ(A ∪ B) ⩽ dimZ A + dimZ B + 1, validity of which was known for metrizable spaces (Rubin [1991]), remains true in general. We also establish the existence of universal spaces of a given cohomological dimension and of a given weight.