complete lattice. If a spot meets a join of others it has to meet some of the constituents, which, roughly speaking, leads to the distributive law where the and are spots and the index family can be arbitrarily large. This distributive law is also satisfied by the lattice of open sets of a topological space. If and are topological spaces with lattices of open sets denoted by and , respectively, and is a continuous map, then, since the pre-image of an open set under a continuous map is open, we obtain a map of lattices in the opposite direction: . Such "opposite-direction" lattice maps thus serve as the proper generalization of continuous maps in the point-free setting. Formal definitions The basic concept is that of a frame, a complete lattice satisfying the general distributive law above; frame homomorphisms are maps between frames that respect all joins (in particular, the least element of the lattice) and finite meets (in particular, the greatest element of the lattice). Frames, together with frame homomorphisms, form a category. The opposite category of the category of frames is known as the category of locales. A locale is thus nothing but a frame; if we consider it as a frame, we will write it as . A locale morphism from the locale to the locale is given by a frame homomorphism . Every topological space gives rise to a frame of open sets and thus to a locale. A locale is called spatial if it isomorphic (in the category of locales) to a locale arising from a topological space in this manner. Examples of locales As mentioned above, every topological space gives rise to a frame of open sets and thus to a locale, by definition a spatial one. Given a topological space , we can also consider the collection of its regular open sets. This is a frame using as join the interior of the closure of the union, and as meet the intersection. We thus obtain another locale associated to . This locale will usually not be spatial. For each and each , use a symbol and construct the free frame on these symbols, modulo the relations (where denotes the greatest element and the smallest element of the frame.) The resulting locale is known as the "locale of surjective functions ". The relations are designed to suggest the interpretation of as the set of all