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, O(X) is a poset, of a rather special kind, and we start by examining when a poset L is 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 can be completed to a sober space. This is explored in Section 8.2. The sober spaces have wonderful properties, and the single most important result about sober spaces is the Hofmann–Mislove Theorem, which we establish and whose consequences we state in Section 8.3. This is the starting point of a theory of correspondences between certain spaces and certain lattices: sober spaces and spatial lattices, but also sober locally compact spaces and continuous distributive complete lattices, or continuous dcpos with their Scott topology and completely distributive lattices notably. We then look at limits and colimits in Top, and how they are preserved or not by the sobrification functor in Section 8.4.