Factorization theorems for cohomological dimension

AbstractThe well-known factorization theorems for covering dimension dim and compact Hausdorff spaces are here established for the cohomological dimension dimZ using a new characterization of dimZ In particular, it is proved that every mapping f: X → Y from a compact Hausdorff space X with dimZ X ⩽ n to a compact metric space Y admits a factorization f = hg, where g: X → Z, h: Z → Y and Z is a metric compactum with dimZ Z ⩽ n. These results are applied to the well-known open problem whether dim X = dimZ X. It is shown that the problem has a positive answer for compact Hausdorff spaces X if and only if it has a positive answer for metric compacta X