Finite spaces and an axiomatization of the Lefschetz number

Abstract

Abstract In [1] Arkowitz and Brown presented an axiomatization of the reduced Lefschetz number of self-maps of finite CW-complexes. By the results of McCord [8], finite simplicial complexes are closely related to finite T 0 {T_{0}} -spaces. This connection and the axioms given by Arkowitz and Brown suggest an axiomatization of the reduced Lefschetz number of maps of finite T 0 {T_{0}} -spaces. However, using the notion of the subdivision of a finite T 0 {T_{0}} -space, we consider the degree and the Lefschetz number of not only self-maps. We also present some properties of the degree of maps between finite models of the circle 𝕊 1 {\mathbb{S}^{1}} .