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:

Equivalent descriptions

Let X be a topological space. Each of the following properties are equivalent to the property of X being spectral:

Properties

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

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

Citations