Existence of Fixed Points for Mappings of Finite Sets

We show that the existence theorem for zeros of a vector field (fixed points of a mapping) holds in the case of a “convex” finite set $X$ and a “continuous” vector field (a self-mapping) directed inwards into the convex hull $\operatorname{co}X$ of $X$. The main goal is to give correct definitions of the notions of “continuity” and “convexity”. We formalize both these notions using a reflexive and symmetric binary relation on $X$, i.e., using a proximity relation. Continuity (we shall say smoothness) is formulated with respect to any proximity relation, and an additional requirement on the proximity (we shall call it the acyclicity condition) transforms $X$ into a “convex” set. If these two requirements are satisfied, then the vector field has a zero (i.e., a fixed point).