Recall that a [door space](https://en.wikipedia.org/wiki/Door_space) is a topological space where every set is either open or closed (or both). A topological space is _finite_ if it has finitely many points. I'm interested in learning about finite door spaces.

**Question 0:** What is an example of a finite topological space which is $T_0$ but not a door space?

**Question 1:** Let $n$ be a natural number. How many door topologies are there on a set with $n$ elements?

I'm interested in both the "labeled" and "unlabeled" versions of question 1.

Recall that via the specialization order, finite topological spaces are equivalent to finite qosets, i.e. quasi-(partially)ordered sets.

**Question 2:** Which finite posets are the specialization order of a door space?