In mathematics, a spectral space is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos.

Definition

Let ''X'' be a topological space and let ''K''^$\circ$(''X'') be the set of all

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...

open subsets of ''X''. Then ''X'' is said to be ''spectral'' if it satisfies all of the following conditions: *''X'' is

and T₀. * ''K''^$\circ$(''X'') is a

basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...

of open subsets of ''X''. * ''K''^$\circ$(''X'') is

closed under In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but ...

finite intersections. * ''X'' is sober, i.e., every nonempty irreducible

closed subset In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...

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 T₀-spaces. #''X'' is homeomorphic to the

spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...

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 In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play a fundamental role in the study of distributi ...

. #''X'' is a T₀ space whose

frame A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent. Frame and FRAME may also refer to: Physical objects In building construction *Framing (co ...

of open sets is coherent (and every coherent frame comes from a unique spectral space in this way).

Properties

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 T₀ by definition, but in general not T₁. In fact a spectral space is T₁ if and only if it is Hausdorff (or T₂) if and only if it is a boolean space if and only if ''K''^$\circ$(''X'') is a

boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...

. *''X'' can be seen as a pairwise Stone space.G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). "Bitopological duality for distributive lattices and Heyting algebras." ''Mathematical Structures in Computer Science'', 20.

Spectral maps

A spectral map ''f: X → Y'' between spectral spaces ''X'' and ''Y'' is a continuous map such that the

preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through ...

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''^$\circ$(''X'').

Citations

References

* M. Hochster (1969). Prime ideal structure in commutative rings. '' Trans. Amer. Math. Soc.'', 142 43—60 *. * {{DEFAULTSORT:Spectral Space General topology Algebraic geometry Lattice theory