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
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—Ralph Waldo Emerson (1803–1882)
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—John Locke (1632–1704)