In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a spectral space is a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 which is homeomorphic to the spectrum of a commutative ring

Spectrum of a ring

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...

.

Definition

Let X be a topological space and let K(X) be the set of all
quasi-compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 and open subsets

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

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

• X is quasi-compact and T₀
  Kolmogorov space
  In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other. This condition, called the T0 condition, is one of the separation axioms...
  
  .
• K(X) is a basis of open subsets of X.
• K(X) is closed under
  Closure (mathematics)
  In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
  
   under finite intersections.
• X is sober
  Sober space
  In mathematics, a sober space is a topological spacesuch that every irreducible closed subset of X is the closure of exactly one point of X: that is, has a unique generic point.-Properties and examples :...
  
  , i.e. every nonempty irreducible
  Hyperconnected space
  In mathematics, a hyperconnected space is a topological space X that cannot be written as the union of two proper closed sets. The name irreducible space is preferred in algebraic geometry....
  
   closed subset
  Closed set
  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...
  
   of X has a (necessarily unique) generic point
  Generic point
  In mathematics, in the fields general topology and particularly of algebraic geometry, a generic point P of a topological space X is an algebraic way of capturing the notion of a generic property: a generic property is a property of the generic point.- Definition and motivation :A generic point of...
  
  .

Equivalent descriptions

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

1. X is homeomorphic to a projective limit of finite T₀-space
   Kolmogorov space
   In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other. This condition, called the T0 condition, is one of the separation axioms...
   
   s.
2. X is homeomorphic to the spectrum
   Duality theory for distributive lattices
   In mathematics, duality theory for distributive lattices provides three different representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces...
   
    of a bounded distributive lattice
   Distributive lattice
   In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
   
    L. In this case, L is isomorphic (as a bounded lattice) to the lattice K(X) (this is called Stone representation of distributive lattices).
3. X is homeomorphic to the spectrum of a commutative ring
   Spectrum of a ring
   In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
   
   .
4. X is the topological space determined by a Priestley space
   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 distributive lattices...
   
   .
5. X is a coherent space
   Coherent space
   In proof theory, a coherent space is a concept introduced in the semantic study of linear logic.Let a set C be given. Two subsets S,T ⊆ C are said to be orthogonal, written S ⊥ T, if S ∩ T is ∅ or a singleton...
   
    in the sense of topology (this indeed is only another name).

Properties

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

• K(X) is a bounded sublattice
  Lattice (order)
  In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
  
   of subsets of X.
• Every closed subspace
  Subspace topology
  In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
  
   of X is spectral.
• An arbitrary intersection of quasi-compact and open subsets of X (hence of elements from K(X)) is again spectral.
• X is T₀
  Kolmogorov space
  In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other. This condition, called the T0 condition, is one of the separation axioms...
  
   by definition, but in general not T₁
  T1 space
  In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
  
  . In fact a spectral space is T₁ if an only if it is Hausdorff
  Hausdorff space
  In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
  
   (or T₂) if and only if it is a boolean space.
• X can be seen as a Pairwise Stone space
  Pairwise Stone space
  In mathematics and particularly in topology, pairwise Stone space is a bitopological space \scriptstyle which is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional....
  
  .

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

Equivalence of categories

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics...

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(X)