Spectral space

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".

External links

* [http://folli.loria.fr/cds/1999/library/pdf/bezhan.pdf see 8.3 - Definition 6 and bibliography]