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

Abstract

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 $$ dim $$ .