TITLE: Example of a non-hausdorff space
QUESTION [19 upvotes]: For instance, the simplest example of a non-hausdorff topological space is the the pair $(X, \tau_X)$ where $\tau_X = \{ X, \varnothing \} $. But this is boring. Can someone help me find more interesting examples?

REPLY [2 votes]: Another topological space, which is not Hausdorff (unfortunately not T1, either), is the open interval $(0,1)$ with the nested interval topology.
Similarly to the cofinite topology (it's T1!) (and verifiable almost analogously) the cocountable Topology
$$
\mathcal{T}
:= \{ U \subset X: U = \emptyset \text{ or } U^C \text{ is countable}\}
$$
of a infinite set $X$ is also not Hausdorff.