The mardešić factorization theorem for extension theory and C-separation

We shall prove a type of Mardešić factorization theorem for extension theory over an arbitrary stratum of CW-complexes in the class of arbitrary compact Hausdorff spaces. Our result provides that the space through which the factorization occurs will have the same strong countability property (e.g., strong countable dimension) as the original one had. Taking into consideration the class of compact Hausdorff spaces, this result extends all previous ones of its type. Our factorization theorem will simultaneously include factorization for weak infinite-dimensionality and for Property C, that is, for C-spaces. A corollary to our result will be that for any weight a and any finitely homotopy dominated CW-complex K, there exists a Hausdorff compactum X with weight wX ≤ α and which is universal for the property X τ K and weight ≤ α. The condition X τ K means that for every closed subset A of X and every map f A → K, there exists a map F X → K which is an extension of f. The universality means that for every compact Hausdorff space Y whose weight is ≤ α and for which Y τ K is true, there is an embedding of Y into X. We shall show, on the other hand, that there exists a CW-complex 5 which is not finitely homotopy dominated but which has the property that for each weight a, there exists a Hausdorff compactum which is universal for the property X τ S and weight ≤ α.