Editing Pointless topology (section)

== The theory of locales ==
We have seen that we have a functor <math>\Omega</math> from the [[Category of topological spaces|category of topological spaces and continuous maps]] to the category of locales. If we restrict this functor to the full subcategory of [[Sober space|sober spaces]], we obtain a [[full embedding]] of the category of sober spaces and continuous maps into the category of locales. In this sense, locales are generalizations of sober spaces.

It is possible to translate most concepts of [[point-set topology]] into the context of locales, and prove analogous theorems. Some important facts of classical topology depending on [[Axiom of choice|choice principles]] become choice-free (that is, [[Constructivism (mathematics)|constructive]], which is, in particular, appealing for computer science). Thus for instance, arbitrary products of [[Compact space|compact]] locales are compact constructively (this is [[Tychonoff's theorem]] in point-set topology), or completions of uniform locales are constructive. This can be useful if one works in a [[topos]] that does not have the axiom of choice.{{sfn|Johnstone|1983}} Other advantages include the much better behaviour of [[Paracompact space|paracompactness]], with arbitrary products of paracompact locales being paracompact, which is not true for paracompact spaces, or the fact that subgroups of localic groups are always closed.

Another point where topology and locale theory diverge strongly is the concepts of subspaces versus sublocales, and density: given any collection of dense sublocales of a locale <math>X</math>, their intersection is also dense in <math>X</math>.<ref>{{Cite book |last=Johnstone |first=Peter T. |title=Sketches of an Elephant |year=2002 |chapter=C1.2 Locales and Spaces}}</ref> This leads to [[John R. Isbell|Isbell]]'s density theorem: every locale has a smallest dense sublocale. These results have no equivalent in the realm of topological spaces.