The definition of a spectral space requires four conditions:
- The space is compact,
- The space is Kolmogorov (or $T_0$),
- The compact open subsets form a basis of the topology and are closed under finite intersection,
- The space is sober, i.e. every nonempty irreducible closed subset has a generic point.
Is there an example of a space satisfying (1), (2) and (3) but not (4)?
A simple example would be appreciated.