Cell-like resolutions in the strongly countable Z-dimensional case

Suppose that X is a nonempty compact metrizable space and X1⊂X2⊂⋯ is a sequence of nonempty closed subspaces such that for each k∈N, dimZXk⩽k<∞. We show that there exists a compact metrizable space Z, having closed subspaces Z1⊂Z2⊂⋯ , and a surjective cell-like map π :Z→X, such that for each k∈N, (a) k, Xk, and Zk→Xk is a cell-like map. ver, there is a sequence A0⊂A1⊂⋯ of closed subspaces of Z such that for each k, Zk⊂Ak, dimAk⩽k, π|Ak :Ak→X is surjective, and for k∈N, π|Ak :Ak→X is a UVk−1-map.