Spectral space

In mathematics, a spectral space (sometimes called a coherent space) is a topological space that is homeomorphic to the spectrum of a commutative ring.


Let X be a topological space and let K(X) be the set of all quasi-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:

  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.


Let X be a spectral space and let K(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 quasi-compact subset of Y under f is again quasi-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)
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