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^{${\displaystyle \circ }$}(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:

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 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). In this anti-equivalence, a spectral space X corresponds to the lattice K^{${\displaystyle \circ }$}(X).

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