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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 finiteT  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 finiteT  0{T_{0}}   -spaces. However, using the notion of the subdivision of a finiteT  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  1{\mathbb{S}^{1}}   .

Finite spaces and an axiomatization of the Lefschetz number