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
Famous quotes containing the words equivalent and/or descriptions:
“Nobody can deny but religion is a comfort to the distressed, a cordial to the sick, and sometimes a restraint on the wicked; therefore whoever would argue or laugh it out of the world without giving some equivalent for it ought to be treated as a common enemy.”
—Mary Wortley, Lady Montagu (1689–1762)
“Our Lamaze instructor . . . assured our class . . . that our cervix muscles would become “naturally numb” as they swelled and stretched, and deep breathing would turn the final explosions of pain into “manageable discomfort.” This descriptions turned out to be as accurate as, say a steward advising passengers aboard the Titanic to prepare for a brisk but bracing swim.”
—Mary Kay Blakely (20th century)