Spectral space

In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topoi.

Definition

Let X be a topological space and let K^$\circ$(X) be the set of all

compact open subsets of X. Then X is said to be spectral if it satisfies all of the following conditions:

X is compact and T₀.
K^$\circ$(X) is a basis of open subsets of X.
K^$\circ$(X) is closed under finite intersections.
X is sober, i.e., every nonempty irreducible closed subset of X has a (necessarily unique) generic point.

Equivalent descriptions

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^$\circ$(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 T₀ space whose locale of open sets is coherent (and every coherent locale comes from a unique spectral space in this way).

Properties

Let X be a spectral space and let K^$\circ$(X) be as before. Then:

K^$\circ$(X) is a bounded sublattice of subsets of X.
Every closed subspace of X is spectral.
An arbitrary intersection of compact and open subsets of X (hence of elements from K^$\circ$(X)) is again spectral.
X is T₀ by definition, but in general not T₁.A.V. Arkhangel'skii, L.S. Pontryagin (Eds.) General Topology I (1990) Springer-Verlag {{isbn|3-540-18178-4}} (See example 21, section 2.6.) 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 if and only if K^$\circ$(X) is a boolean algebra.
X can be seen as a pairwise Stone space.G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). "Bitopological duality for distributive lattices and Heyting algebras." Mathematical Structures in Computer Science, 20.

Spectral maps

A spectral map f: X → Y between spectral spaces X and Y is a continuous map such that the preimage of every open and compact subset of Y under f is again compact.

The category of spectral spaces, which has spectral maps as morphisms, is dually equivalent to the category of bounded distributive lattices (together with homomorphisms of such lattices).{{sfn|Johnstone|1982}} In this anti-equivalence, a spectral space X corresponds to the lattice K^$\circ$(X).

Spectral space

Definition

Equivalent descriptions

Properties

Spectral maps

References

Further reading