Order- and graph-theoretic investigation of dimensions of finite topological spaces and Alexandroff spaces

摘要

Using concepts from order theory and graph theory we investigate the dimensions ind, Ind and dim of finite topological spaces and Alexandroff spaces. We present specifications of them by means of specialisation pre-orders and algorithms for their computation. For finite spaces we give sharp upper bounds, characterisations of maximal-dimensional spaces via specialisation pre-orders and determine the number of maximal-dimensional spaces on a given set and whether these spaces are homeomorphic. These questions are also investigated for zero-dimensional Alexandroff spaces. We also consider relationships between the dimensions ind, Ind and dim.