Door not implies discrete

Statement
It is possible for a topological space to be a door space (i.e., every subset is either an open subset or a closed subset) but not a discrete space.

Proof
Consider the Sierpiński space: a set $$X$$ with two distinct elements $$a,b$$. Define the open subsets of $$X$$ as the subsets:

$$\{ \}, \{ a \}, \{ a,b \}$$

Correspondingly, the closed subsets are:

$$\{ \}, \{ b \}, \{ a,b \}$$

This is a door space, because every subset is either open or closed (and some are both). On the other hand, it is not discrete because not every subset is open.