Editing Spectral space

{{Short description|Homeomorphic topological space}}
In [[mathematics]], a '''spectral space''' is a [[topological space]] that is [[homeomorphic]] to the [[Spectrum of a ring|spectrum of a commutative ring]]. It is sometimes also called a '''coherent space''' because of the connection to [[coherent topos|coherent topoi]].

==Definition==

Let ''X'' be a topological space and let ''K''<sup><math>\circ</math></sup>(''X'') be the set of all 
[[Compact space|compact]] [[Open set|open subsets]] of ''X''. Then ''X'' is said to be ''spectral'' if it satisfies all of the following conditions:
*''X'' is [[compact space|compact]] and [[Kolmogorov space|T<sub>0</sub>]].
* ''K''<sup><math>\circ</math></sup>(''X'') is a [[basis (topology)|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 '''[[duality theory for distributive lattices|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 T<sub>0</sub> space whose [[Frames and locales|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''<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 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>]].<ref>[[Alexander Arhangelskii|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.)''</ref> 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]] if and only if ''K''<sup><math>\circ</math></sup>(''X'') is a [[boolean algebra]].
*''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 compact subset of ''Y'' under ''f'' is  again compact.

The [[category (mathematics)|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 [[homomorphism]]s of such lattices).{{sfn|Johnstone|1982}} In this anti-equivalence, a spectral space ''X'' corresponds to the lattice ''K''<sup><math>\circ</math></sup>(''X'').

==References==
{{reflist}}

==Further reading==
{{refbegin}}
*[[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}}.
* {{cite book | last1=Dickmann | first1=Max | last2=Schwartz | first2= Niels | last3=Tressl | first3= Marcus | title=Spectral Spaces| doi=10.1017/9781316543870 | year=2019 | publisher=[[Cambridge University Press]] | series=New Mathematical Monographs | volume=35 | location=Cambridge | isbn=9781107146723 }}
{{refend}}

{{DEFAULTSORT:Spectral Space}}
[[Category:General topology]]
[[Category:Algebraic geometry]]
[[Category:Lattice theory]]