Connected door spaces and topological solutions of equations

TL;DR

This paper classifies connected door topological spaces and demonstrates their relation to solutions of specific valuation equations, providing new insights into their structure and solutions.

Contribution

It offers an elementary proof of the classification theorem and links connected door spaces to valuation equations, including special solutions as unions of such spaces.

Findings

Connected door spaces are classified.

Connected door topologies solve specific valuation equations.

Special solutions are constructed as unions of connected door spaces.

Abstract

The connected door space is an enigmatic topological space in which every proper nonempty subset is either open or closed, but not both. This paper provides an elementary proof of the classification theorem of connected door spaces. More importantly, we show that connected door topologies can be viewed as solutions of the valuation $f(A)+f(B)=f(A\cup B)+f(A\cap B)$ and the equation $f(A)+f(B)=f(A\cup B)$, respectively. In addition, some special solutions, which can be regarded as a union of connected door spaces, are provided.