Dimension in a reasonable class of spaces

Let P be a class of compact spaces satisfying:(a)P is closed with respect to taking finite products.(b)P is closed with respect to taking closed subspaces.(c)P contains all metrizable compacta.(d)Every member of P can be mapped onto a metrizable space by a zero-dimensional map. Seven distinct dimension functions coincide on completely paracompact subspaces of members of P, and each subspace of a member of P has a compactification in P of the same weight and dimension. Examples of classes satisfying (a), (b), (c) and (d) are several classes of Eberlein compactifications of metrizable spaces. There are, however, uniform Eberlein compacta with dim