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Concept# Spectral space

Summary

In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos.
Definition
Let X be a topological space and let K\circ(X) be the set of all
compact open subsets of X. Then X is said to be spectral if it satisfies all of the following conditions:
*X is compact and T0.

- K\circ(X) is a basis of open subsets of X.
- K\circ(X) is closed under finite intersections.
- X is sober, i.e., every nonempty irreducible closed subset of X has a (necessarily unique) generic point.

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