Spectral space

(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.

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 homomorphisms of such lattices).

[3] In this anti-equivalence, a spectral space X corresponds to the lattice K