Spectral space

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In mathematics, a topological space X with topology Ω is said to be spectral if

• 1) X is compact and T₀;
• 2) The set C(X) of all compact-open subsets of (X,Ω) is a sublattice of Ω and a base for the topology. Note that "compact-open" does not mean a set that is both compact and open: such a set would be unusual in most interesting spaces. Here, it means locally compact. That is, a compact-open set is an open set whose closure is compact, where the closure of a set A is the intersection of all the closed sets that contain A;
• 3) X is sober, that is any nonempty closed set F which is not a closure of a singleton {x} is a union of two closed sets which differ from F.

[edit] External links

• see 8.3 - Definition 6 and bibliography