Refinable maps in dimension theory
A map f from a compact metric space X onto a compact metric space Y is said to be refinable if for each \(\epsilon>0\) there is an \(\epsilon\)-map from X onto Y whose distance from f is less than \(\epsilon\). The notion of refinability can be extended to maps between arbitrary spaces. It is proved that refinable maps between compact Hausdorff spaces preserve covering dimension exactly and also preserve weak infinite-dimension. A more specialized notion of a closed-refinable (c-refinable) map is introduced. It is shown that if a surjective map between normal spaces is c-refinable, then its Stone-Čech extension is refinable, from which it is deduced that c-refinable maps between normal spaces preserve covering dimension exactly. The paper concludes with a discussion of the relationship between the classes of c-refinable and proper refinable mappings.
scientific article; zbMATH DE number 3859992