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 topoi.

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

*X*is compact and T_{0}.*K*^{$\circ$}(*X*) is a basis of open subsets of*X*.*K*^{$\circ$}(*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 T_{0}-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*^{$\circ$}(*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 T_{0}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*^{$\circ$}(*X*) be as before. Then:

*K*^{$\circ$}(*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*^{$\circ$}(*X*)) is again spectral. *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 if and only if*K*^{$\circ$}(*X*) is a boolean algebra.*X*can be seen as a pairwise Stone space.^{[2]}

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*^{$\circ$}(*X*).

**^**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.

- 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. - Dickmann, Max; Schwartz, Niels; Tressl, Marcus (2019).
*Spectral Spaces*. New Mathematical Monographs. Vol. 35. Cambridge: Cambridge University Press. doi:10.1017/9781316543870. ISBN 9781107146723.