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. 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 ∘ {displaystyle circ } (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: Let X be a topological space. Each of the following properties are equivalent to the property of X being spectral: Let X be a spectral space and let K ∘ {displaystyle circ } (X) 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 (together with morphisms of such lattices). In this anti-equivalence, a spectral space X corresponds to the lattice K ∘ {displaystyle circ } (X).