spectral space

A topological space is called spectral if

• it is compact,

• Kolmogorov (also called $T_0$ (http://planetmath.org/T0)),

• compactness is preserved upon finite intersection of open compact sets, and

• any nonempty irreducible subspace of it contains a generic point

In his thesis, Mel Hochster showed that for any spectral space there is commutative unitary ring whose prime spectrum is homeomorphic to the spectral space.