A topological space $X$ is a door space if any subset of $X$ is either open or closed (or both). Naturally, a connected door space is that in which any proper subset is either open or closed, but not both. According to this paper, there are only three types of topologies yielded by connected door spaces: particular point topologies, excluded point topologies, and $T_1$ topologies in which any two non-disjoint open sets have infinite intersection.
Is there any explicit example of a connected door space of the third type? The cofinite topology on any infinite set satisfies the third type but is never a door space, and I can't really think of spaces in which open intersections are infinite...
Let $X$ be an infinite set and (using a mild form of choice) $\mathcal{F}$ a free ultrafilter on $X$.
Let $\mathcal{T}=\{\emptyset\} \cup \mathcal{F}$, which is a topology on $X$: unions follow from closedness under enlargements, finite intersections follow from the filter axioms as well, and we added the empty set explicitly. That any two non-empty open sets intersect also is a filter axiom ($\emptyset \notin \mathcal{F})$.
It is well known that for all subsets $A$ of $X$ either $A$ or its complement is in $\mathcal{F}$ (this is a classic ultrafilter property). This both implies that $X$ is $T_1$ ( as the ultrafilter is free it contains all cofinite sets) and that it is a door space.
This construction gives us a lot of non-homeomorphic spaces, as there are many so-called types of ultrafilters; See set theory for more on this.