  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.
