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In [[mathematics]], a '''spectral space''' is a [[topological space]] which is [[homeomorphic]] to the [[Spectrum of a ring|spectrum of a commutative ring]].
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==Definition==
 
Let ''X'' be a topological space and let ''K<sup><math>\circ</math></sup>(X)'' be the set of all
[[Compact space|quasi-compact]] [[Open set|open subsets]] of ''X''. Then ''X'' is said to be ''spectral'' if it satisfies all of the following conditions:
*''X'' is [[quasi-compact]] and [[Kolmogorov space|''T<sub>0</sub>'']].
* ''K<sup><math>\circ</math></sup>(X)'' is a basis of open subsets of ''X''.
* ''K<sup><math>\circ</math></sup>(X)'' is [[Closure (mathematics)|closed under]] finite intersections.
* ''X'' is [[Sober space|sober]], i.e. every nonempty [[Hyperconnected space|irreducible]] [[Closed set|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 [[Kolmogorov space|''T<sub>0</sub>''-space]]s.
#''X'' is homeomorphic to the [[duality theory for distributive lattices|spectrum]] of a [[distributive lattice|bounded distributive lattice]] ''L''. In this case, ''L'' is isomorphic (as a bounded lattice) to the lattice ''K<sup><math>\circ</math></sup>(X)'' (this is called '''Stone representation of distributive lattices''').
#''X'' is homeomorphic to the [[Spectrum of a ring|spectrum of a commutative ring]].
#''X'' is the topological space determined by a [[Priestley space]].
#''X'' is a [[coherent space]] in the sense of topology (this indeed is only another name).
 
==Properties==
 
Let ''X'' be a spectral space and let ''K<sup><math>\circ</math></sup>(X)'' be as before. Then:
*''K<sup><math>\circ</math></sup>(X)'' is a [[Lattice (order)|bounded sublattice]] of subsets of ''X''.
*Every closed [[Subspace topology|subspace]] of ''X'' is spectral.
*An arbitrary intersection of quasi-compact and open subsets of ''X'' (hence of elements from ''K<sup><math>\circ</math></sup>(X)'') is again spectral.
*''X'' is [[Kolmogorov space|T<sub>0</sub>]] by definition, but in general not [[T1 space|T<sub>1</sub>]]. In fact a spectral space is T<sub>1</sub> if and only if it is [[Hausdorff space|Hausdorff]] (or T<sub>2</sub>) if and only if it is a [[boolean space]].
*''X'' can be seen as a [[Pairwise Stone space]].<ref>G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). Bitopological duality for distributive lattices and Heyting algebras. ''Mathematical Structures in Computer Science'', 20.</ref>
 
==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 [[Equivalence of categories|dually equivalent]] to the category of bounded distributive lattices (together with morphisms of such lattices).<ref>{{harv|Johnstone|1982}}</ref> In this anti-equivalence, a spectral space ''X'' corresponds to the lattice K<sup><math>\circ</math></sup>(X).
 
==References==
 
*[[Mel Hochster|M. Hochster]] (1969). Prime ideal structure in commutative rings. ''Trans. Amer. Math. Soc.'', 142 43—60
 
*{{citation
| last = Johnstone | first = Peter | author-link = Peter Johnstone (mathematician)
| isbn = 978-0-521-33779-3
| publisher = Cambridge University Press
| title = Stone Spaces
| contribution = II.3 Coherent locales
| pages = 62–69
| year = 1982}}.
 
==Footnotes==
 
{{reflist}}
 
{{DEFAULTSORT:Spectral Space}}
[[Category:General topology]]
[[Category:Algebraic geometry]]
[[Category:Lattice theory]]

Latest revision as of 01:03, 15 September 2014

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