On functors preserving skeletal maps and skeletally generated compacta

TL;DR

This paper characterizes when normal functors on compact spaces preserve skeletal maps and shows they also preserve skeletally generated compacta, revealing differences from open functors.

Contribution

It provides a characterization of skeletal functors via open maps and proves that normal functors preserve skeletally generated compacta, contrasting with known results about open functors.

Findings

Normal functors are skeletal iff they preserve skeletal epimorphisms for specific open maps.

Open normal functors are skeletal.

Normal functors preserve 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.