Spectral space

Summary

A spectral space is a compact sober topological space whose compact open sets are closed under finite intersections and form a basis of the topology. The prime spectrum of any commutative ring with the Zariski topology is spectral, and by Hochster's theorem every spectral space arises in this way. The compact open sets of a spectral space form a bounded distributive lattice, and every bounded distributive lattice arises in this way. So spectral spaces connect commutative algebra, lattice theory and topology.