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 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.K is a basis of open subsets of X.K is closed under finite intersections.X is sober, i.e., every nonempty irreducible closed subset of X has a unique generic point.

From that X is sober it follows that X is T₀. Indeed the definition of a spectral space can be equivalently reformulated through explicitly assuming that X is T₀ and weaking the assumption that X is sober to only require it to be quasi-sober, i.e. every irreducible closed subspace possesses a generic point. This is the way the definition is formulated in Hochster's 1967 thesis.

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 duality theory for [distributive lattices|spectrum] of a bounded distributive lattice L. In this case, L is isomorphic to the lattice K.
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.

Properties

Let X be a spectral space and let K be as before. Then:

K 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 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 if and only if it is a boolean space if and only if K is a boolean algebra.
X can be seen as a pairwise Stone space.

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. In this anti-equivalence, a spectral space X corresponds to the lattice K.