- X is quasi-compact and T0.
- K(X) is a basis of open subsets of X.
- K(X) is closed under finite intersections.
- X is sober, i.e. every nonempty irreducible closed subset of X has a (necessarily unique) generic point.
Let X be a topological space. Each of the following properties are equivalent to the property of X being spectral:
- X is homeomorphic to a projective limit of finite T0-spaces.
- X is homeomorphic to the spectrum of a bounded distributive lattice L. In this case, L is isomorphic (as a bounded lattice) to the lattice K(X) (this is called Stone representation of distributive lattices).
- X is homeomorphic to the spectrum of a commutative ring.
- X is the topological space determined by a Priestley space.
- X is a coherent space in the sense of topology (this indeed is only another name).
Let X be a spectral space and let K(X) be as before. Then:
- K(X) is a bounded sublattice of subsets of X.
- Every closed subspace of X is spectral.
- An arbitrary intersection of quasi-compact and open subsets of X (hence of elements from K(X)) is again spectral.
- X is T0 by definition, but in general not T1. In fact a spectral space is T1 if and only if it is Hausdorff (or T2) if and only if it is a boolean space.
- X can be seen as a Pairwise Stone space.
A spectral map f: X → Y between spectral spaces X and Y is a continuous map such that the preimage of every open and quasi-compact subset of Y under f is again quasi-compact.
The category of spectral spaces which has spectral maps as morphisms is dually equivalent to the category of bounded distributive lattices (together with morphisms of such lattices). In this anti-equivalence, a spectral space X corresponds to the lattice K(X).
- M. Hochster (1969). Prime ideal structure in commutative rings. Trans. Amer. Math. Soc., 142 43—60
- Johnstone, Peter (1982), "II.3 Coherent locales", Stone Spaces, Cambridge University Press, pp. 62–69, ISBN 978-0-521-33779-3.
- G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). Bitopological duality for distributive lattices and Heyting algebras. Mathematical Structures in Computer Science, 20.
- (Johnstone 1982)