On monomorphic topological functors with finite supports

Abstract

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