Editing Spectral space (section)

==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).