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 T0-spaces.
- X is homeomorphic to the spectrum of a bounded distributive lattice L. In this case, L is isomorphic (as a bounded lattice) to the lattice K(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 coherent space in the sense of topology (this indeed is only another name).
Read more about this topic: Spectral Space
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