Application of the Different Topology Models on a Finite Set to the Enumeration and Calculation of All Labeled Т0-Topologies with a Given Weight

EasyChair Preprint 11452

Abstract

The tasks of studying topologies on a finite set are closely related to the tasks of studying bijunctive Boolean functions, partial orders, and graphs. This explains the possibility of using combinatorial methods for their research, and also makes such topologies important mathematical objects.

The problem of enumeration of homotopy types of finite topological spaces is as important as the problem of enumeration of homotopy types of finite simplicial complexes. These tasks are related to the digital processing of images based on finite sets of observations, that is, trying to understand the content of the image based on the concept of proximity of points.

The problem of computing topologies on a finite set remains unsolved at the moment, so interest in it remains, publications by researchers from different countries with new results have appeared.

This paper uses two models to study topology on a finite set. In the first model, each topology is assigned a nondecreasing sequence of non-negative integers (topology vector), in the second - the conjunctive normal form of the Boolean function.

Keyphrases: T0-topologies, consistent topologies., minimum element neighborhood, topology vector, topology weight

@booklet{EasyChair:11452,
  author    = {Anna Skryabina and Polina Stegantseva},
  title     = {Application of the Different Topology Models on a Finite Set to the Enumeration and Calculation of All Labeled Т0-Topologies with a Given Weight},
  howpublished = {EasyChair Preprint 11452},
  year      = {EasyChair, 2023}}