# Example of a compact, Kolmogorov space with a basis of compact open sets, but not spectral.

The definition of a spectral space requires four conditions:

1. The space is compact,
2. The space is Kolmogorov (or $T_0$),
3. The compact open subsets form a basis of the topology and are closed under finite intersection,
4. 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.

• This is not quite the definition of a spectral space: in a spectral space, the intersection of two quasi-compact opens must be quasi-compact. Note that my example has this additional property. Jan 18, 2016 at 23:13

The simplest example is an infinite set $X$ equipped with the cofinite topology.
This satisfies the first three conditions, but the whole space $X$ is an irreducible closed set without a generic point.
This also provides a good example of soberification: we can make $X$ into a sober space by adding a single dense point.