On functors preserving skeletal maps and skeletally generated compacta

Abstract

A map $f:X\to Y$ between topological spaces is skeletal if the preimage $f^{-1}(A)$ of each nowhere dense subset $A\subset Y$ is nowhere dense in $X$. We prove that a normal functor $F:Comp\to Comp$ is skeletal (which means that $F$ preserves skeletal epimorphisms) if and only if for any open surjective open map $f:X\to Y$ between zero-dimensional compacta with two-element non-degeneracy set $N^f=\{x\in X:|f^{-1}(f(x))|>1\}$ the map $Ff:FX\to FY$ 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 $F: Comp\to Comp$ preserves the class of skeletally generated compacta. This contrasts with the known Shchepin's result saying that a normal functor is open if and only if it preserves openly generated compacta.