Computational methods in topology and dynamical systems

Abstract

Alexandroff spaces are topological spaces satisfying that the arbitrary intersection of open sets is an open set. The theory of Alexandroff spaces is becoming a significant part of topology since it can be used to model and solve mathematical problems of different nature. They were first considered in [1] under the name of diskrete R¨aume (discrete spaces), where P.S. Alexandroff proved that they can also be treated as combinatorial objects. Concretely, he proved that Alexandroff spaces are in bijective correspondence with preordered sets. Therefore, topological notions such as continuity or homotopy can be expressed in terms of combinatorial notions. However, Alexandroff spaces have some poor topological properties. For instance, if X is an Alexandroff space being T1 or a Fr´echet space, then its topology is the discrete topology...