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.


Let X be a topological space and let K (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 descriptionsEdit

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 (as a bounded lattice) to the lattice K (X) (this is called Stone representation of distributive lattices).
  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 (and every coherent frame comes from a unique spectral space in this way).


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 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.[1] 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 (X) is a boolean algebra.
  • X can be seen as a pairwise Stone space.[2]

Spectral mapsEdit

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 morphisms of such lattices).[3] In this anti-equivalence, a spectral space X corresponds to the lattice K (X).


  1. ^ 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.)
  2. ^ G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). "Bitopological duality for distributive lattices and Heyting algebras." Mathematical Structures in Computer Science, 20.
  3. ^ Johnstone 1982.