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

  • Rudolf Berghammer
  • Michael WinterEmail author


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 \( dim \).


Topological space Alexandroff space Dimension Specialisation pre-order Linear order Flat order Hasse-diagram Linear directed binary tree 

Mathematics Subject Classification

54F45 06A06 



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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für InformatikChristian-Albrechts-Universität zu KielKielGermany
  2. 2.Department of Computer ScienceBrock UniversitySaint CatharinesCanada

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