The Euler Characteristic of Finite Topological Spaces

The purpose of this paper is to illustrate the relationship between the topological property of the Euler characteristic and a combinatorial object, the Möbius function, in the context of finite T0-spaces. To do this I first explain the fundamental connection between such spaces and finite partially ordered sets by proving some facts fundamental to the study of finite spaces. Then I define the Euler characteristic and provide some elementary facts pertaining to the Euler characteristic of finite T0-spaces. Finally, I introduce the Möbius function and prove its relationship to the Euler characteristic.