Coincidence theorems for finite topological spaces

Abstract

We adapt the definition of the Vietoris map to the framework of finite topological spaces and we prove some coincidence theorems. From them, we deduce a Lefschetz fixed point theorem for multivalued maps that improves recent results in the literature. Finally, it is given an application to the approximation of discrete dynamical systems in polyhedra.

A combinatorial description of shape theory

“…Because of the need of finding computational methods to study dynamical systems, the theory of finite topological spaces [29,4] has recently grown up in this direction. Classical topological methods in dynamical systems, the Conley index [13,43,35] or the Lefschetz fixed point theorem [26], have been adapted to this framework [27,5,9]. This combinatorial approach has lead to a sort of persistence algorithms that can be used to analyze data collected from dynamical systems [16,17,18].…”

Shape of compacta as extension of weak homotopy of finite spaces

“…Recent research recognizes the critical role played by the theory of finite topological spaces in several fields of mathematics such as dynamical systems [6,16,9], group theory [5,4,11] algebraic topology (see [3,18] and the references given there) and geometric topology [21,10]. It is worth pointing out that important conjectures can be stated in terms of the theory of finite topological spaces.…”