# 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.

## Definition

Let *X* be a topological space and let *K ^{}(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:

*X*is quasi-compact and*T*._{0}-
*K*is a basis of open subsets of^{}(X)*X*. -
*K*is closed under finite intersections.^{}(X) -
*X*is sober, i.e. every nonempty irreducible closed subset of*X*has a (necessarily unique) generic point.

## Equivalent descriptions

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*T*-spaces._{0}*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*(this is called^{}(X)**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.

## Properties

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

*K*is a bounded sublattice of subsets of^{}(X)*X*.- Every closed subspace of
*X*is spectral. - An arbitrary intersection of quasi-compact and open subsets of
*X*(hence of elements from*K*) is again spectral.^{}(X) *X*is T_{0}by definition, but in general not T_{1}.^{[1]}In fact a spectral space is T_{1}if and only if it is Hausdorff (or T_{2}) if and only if it is a boolean space.*X*can be seen as a Pairwise Stone space.^{[2]}

## Spectral maps

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).^{[3]} In this anti-equivalence, a spectral space *X* corresponds to the lattice K^{}(X).

## References

- M. Hochster (1969). Prime ideal structure in commutative rings.
*Trans. Amer. Math. Soc.*, 142 43—60 - Johnstone, Peter (1982), "II.3 Coherent locales",
*Stone Spaces*, Cambridge University Press, pp. 62–69, ISBN 978-0-521-33779-3.

## Footnotes

- ↑ 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.)* - ↑ G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). "Bitopological duality for distributive lattices and Heyting algebras."
*Mathematical Structures in Computer Science*, 20. - ↑ (Johnstone 1982)