Editing Pointless topology (section)

== Examples of locales ==

* As mentioned above, every topological space <math>T</math> gives rise to a frame <math>\Omega(T)</math> of open sets and thus to a locale, by definition a spatial one.
* Given a topological space <math>T</math>, we can also consider the collection of its [[Regular open set|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 <math>T</math>. This locale will usually not be spatial.
* For each <math>n\in\N</math> and each <math>a\in\R</math>, use a symbol <math>U_{n,a}</math> and construct the free frame on these symbols, modulo the relations

::<math>\bigvee_{a\in\R} U_{n,a}=\top   \ \text{  for every }n\in\N</math>

::<math>U_{n,a}\and U_{n,b}=\bot   \ \text{  for every }n\in\N\text{ and all }a,b\in\R\text{ with } a\ne b</math>

::<math>\bigvee_{n\in\N} U_{n,a}=\top   \ \text{  for every }a\in\R</math>

:(where <math>\top</math> denotes the greatest element and <math>\bot</math> the smallest element of the frame.) The resulting locale is known as the "locale of surjective functions <math>\N\to\R</math>". The relations are designed to suggest the interpretation of <math>U_{n,a}</math> as the set of all those surjective functions <math>f:\N\to\R</math> with <math>f(n)=a</math>. Of course, there are no such surjective functions <math>\N\to\R</math>, and this is not a spatial locale.