How to find finite topological spaces with given quotient-spaces

TL;DR

This paper addresses the problem of identifying finite topological spaces based on their quotient-spaces, presenting an algorithm to detect when reconstruction is possible or not, and handling various cases.

Contribution

It introduces an algorithm that determines the existence and reconstructability of finite topological spaces from given quotient-spaces, accounting for non-uniqueness and impossibility cases.

Findings

The algorithm can detect when reconstruction is impossible.

It handles cases with multiple non-homeomorphic spaces sharing quotient-spaces.

The method identifies all situations where reconstruction is feasible or not.

Abstract

Our main problem is to find finite topological spaces to within homeomorphism, given (also to within homeomorphism) the quotient-spaces obtained by identifying one point of the space with each one of the other points. In a previous version of this paper, our aim was to reconstruct a topological space from its quotient-spaces; but a reconstruction is not always possible either in the sense that several non-homeomorphic topological spaces yield the same quotient-spaces, or in the sense that no topological space yields an arbitrarily given family of quotient-spaces. In this version of the paper we present an algorithm that detects, and deals with, all these situations.