On monomorphic topological functors with finite supports

TL;DR

This paper investigates properties of monomorphic topological functors with finite supports, showing they are epimorphic, continuous, and preserve intersections, with implications for their behavior on finite-dimensional ANRs, extending prior results.

Contribution

It proves that such functors are epimorphic, continuous, and preserve intersections, and extends known results on their action on finite-dimensional ANRs.

Findings

Monomorphic functors with finite supports are epimorphic and continuous.

These functors preserve intersections and finite-dimensional ANRs under certain conditions.

The results extend and improve upon Basmanov's earlier work.

Abstract

We prove that a monomorphic functor $F:Comp\to Comp$ with finite supports is epimorphic, continuous, and its maximal $\emptyset$-modification $F^\circ$ preserves intersections. This implies that a monomorphic functor $F:Comp\to Comp$ of finite degree $deg F\le n$ preserves (finite-dimensional) compact ANR's if the spaces $F\emptyset$, $F^\circ\emptyset$, and $Fn$ are finite-dimensional ANR's. This improves a known result of Basmanov.