Spectral space

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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.

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Definition

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 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.
  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).

Properties

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

Spectral maps

A spectral mapf: 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).

Citations

  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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