Connected door spaces and topological solutions of equations

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.