On functors preserving skeletal maps and skeletally generated compacta

A map f : X → Y between topological spaces is skeletal if the preimage f − 1 ( A ) of each nowhere dense subset A ⊂ Y is nowhere dense in X. We prove that a normal functor F : Comp → Comp is skeletal (which means that F preserves skeletal epimorphisms) if and only if for any open surjective map f : X → Y between metrizable zero-dimensional compacta with two-element non-degeneracy set N f = { x ∈ X : | f − 1 ( f ( x ) ) | > 1 } the map F f : F X → F Y is skeletal. This characterization implies that each open normal functor is skeletal. The converse is not true even for normal functors of finite degree. The other main result of the paper says that each normal functor F : Comp → Comp preserves the class of skeletally generated compacta. This contrasts with the known Ščepinʼs result saying that a normal functor is open if and only if it preserves the class of openly generated compacta.