Properties
Let X be a spectral space and let K(X) be as before. Then:
- K(X) is a bounded sublattice of subsets of X.
- Every closed subspace of X is spectral.
- An arbitrary intersection of quasi-compact and open subsets of X (hence of elements from K(X)) is again spectral.
- X is T0 by definition, but in general not T1. In fact a spectral space is T1 if and only if it is Hausdorff (or T2) if and only if it is a boolean space.
- X can be seen as a Pairwise Stone space.
Read more about this topic: Spectral Space
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)