Spectral space

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In mathematics, a spectral space (or coherent space) is a topological space which, informally speaking, resembles the spectrum of a ring with its Zariski topology.

Let X be a topological space and C(X) the collection of all compact open subsets of X, then X is said to be spectral if it satisfies the following conditions:

• X is sober, i.e. every irreducible closed subset of X has a unique generic point;
• C(X) is a base for the topology of X and is closed under finite intersections.

[edit] Properties

Let X be a spectral space and C(X) as before, then:

• X is compact since it is the empty intersection in C(X).
• C(X) is a bounded sublattice of the topology of X.
• Every closed subspace of X is spectral.
• Every compact open subspace of X is spectral.
• Since X is sober it is T₀, but usually not T₁.

[edit] External links

• Peter T. Johnstone, Stone Spaces, section 3.4 on page 65
• Nick Bezhanishvili, Distributive Lattices with Quantifier: Topological Representation, definition 6