Concept

Spectral space

Résumé
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: X is compact and T0. K(X) is a basis of open subsets of X. K(X) is closed under finite intersections. X is sober, i.e., every nonempty irreducible closed subset of X has a (necessarily unique) generic point. Let X be a topological space. Each of the following properties are equivalent to the property of X being spectral: X is homeomorphic to a projective limit of finite T0-spaces. 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). X is homeomorphic to the spectrum of a commutative ring. X is the topological space determined by a Priestley space. 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. 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. A spectral map f: X → Y between spectral spaces X and Y is a continuous map such that the 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).
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