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 topos.

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:
Let X be a topological space. Each of the following properties are equivalent
to the property of X being spectral:
  1. X is homeomorphic to a projective limit of finite T0-spaces.
  2. X is homeomorphic to the spectrum of a bounded distributive lattice L. In this case, L is isomorphic to the lattice K.
  3. X is homeomorphic to the spectrum of a commutative ring.
  4. X is the topological space determined by a Priestley space.
  5. X is a T0 space whose frame of open sets is coherent.

    Properties

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

Footnotes