mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a spectral space is a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

that is

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

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

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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

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

of ''X'' has a (necessarily unique)

generic point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is a point in a ''general position'', at which all generic property, generic properties are true, a generic property being a property which is true for Almost everywhere, ...

Equivalent descriptions

Let ''X'' be a topological space. Each of the following properties are equivalent to the property of ''X'' being spectral: #''X'' is

to a projective limit of finite T₀-spaces. #''X'' is homeomorphic to the

spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

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 T₀ space whose locale of open sets is coherent (and every coherent locale 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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...

. *''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 In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

such that the

preimage In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each ...

of every open and compact subset of ''Y'' under ''f'' is again compact. The

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

of spectral spaces, which has spectral maps as morphisms, is dually equivalent to the category of bounded distributive lattices (together with

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

s of such lattices). In this anti-equivalence, a spectral space ''X'' corresponds to the lattice ''K''^$\circ$(''X'').

Definition

Equivalent descriptions

Properties

Spectral maps

References

Further reading