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Non-Hausdorff Topology and Domain Theory by Jean Goubault-Larrecq

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8

Sober spaces

In Chapter 4, we mentioned a connection between topology and order, centered around the notion of specialization quasi-ordering. The connection between topology and order goes much beyond this. For any topological space X, image(X) is a poset, of a rather special kind, and we start by examining when a poset Lis of this form for some space X. The posets that have this property are the spatial lattices, and Stone duality is a canonical way of retrieving the space X of points from the lattice L alone; see Section 8.1. Conversely, the spaces X of points obtained from spatial lattices are exactly the sober topological spaces, and any space ...

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