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:
“I started off rapping for people just like myself, people who were in awe of wealth and flash. It was a conversation between me and them. But now most of those who buy my records are listening in on others’ conversation. They are the aural equivalent of voyeurs, thrilled at this crazy world that has nothing to do with their experience.”
—Ice-T [Tracy Marrow], U.S. rap musician. Observer (London, Oct. 27, 1991)
“Matter-of-fact descriptions make the improbable seem real.”
—Mason Cooley (b. 1927)