Editing Pointless topology (section)

==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 [[Join (mathematics)|joins]] (in particular, the [[least element]] of the lattice) and finite [[meet (mathematics)|meet]]s (in particular, the [[greatest element]] of the lattice). Frames, together with frame homomorphisms, form a [[category (mathematics)|category]].

The [[opposite category]] of the category of frames is known as the '''category of locales'''. A locale <math>X</math> is thus nothing but a frame; if we consider it as a frame, we will write it as <math>O(X)</math>. A '''locale morphism''' <math>X\to Y</math> from the locale <math>X</math> to the locale <math>Y</math> is given by a frame homomorphism <math>O(Y)\to O(X)</math>.

Every topological space <math>T</math> gives rise to a frame <math>\Omega(T)</math> 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.