nLab
topological locale

Idea

A topological (or spatial) locale is a locale that comes from a topological space. This is an extra property of locales, a property of having enough points.

Definitions

Let $X$ be a topological space. Then we may define a locale, denoted $\Omega (X)$, whose frame of opens is precisely the frame of open subspaces of $X$.

A locale is topological, or spatial, if it is isomorphic to $\Omega (X)$ for some topological space $X$.

A locale $L$ has enough points if, given any two opens $U$ and $V$ in $L$, $U=V$ if (hence iff) precisely the same points of $L$ belong to $U$ as belong to $V$.

Properties

The following conditions are all logically equivalent on a locale $L$:

1. $L$ is topological, as defined above.
2. $L$ has enough points, as defined above.
3. Given any two opens $U$ and $V$ in $L$, $U\le V$ if (hence iff) every point of $L$ that belongs to $U$ also belongs to $V$.
4. $L$ is isomorphic to $\Omega (\mathrm{pt}(L))$, where $\mathrm{pt}(L)$ is the space of points? of $L$.
5. The natural morphism ${\eta }_L:\Omega (\mathrm{pt}(L))\to L$ (the counit of the adjunction from Top and Loc) is an isomorphism.

(It would be nice to state this as a theorem and put in a proof.)

Basically, what is going on here is that we have an idempotent adjunction from topological spaces to locales, and the topological locales comprise the image of this adjunction. The corresponding condition on topological spaces is being sober.

Therefore, the full subcategory of $\mathrm{Loc}$ on the topological locales is equivalent to the full subcategory of $\mathrm{Top}$ on sober spaces.

Terminology

The terms ‘topological locale’ and ‘spatial locale’ can be confusing; they suggest a locale in Top or in some category Sp? of spaces, which is not correct. Instead, the adjective ‘topological’ and ‘spatial’ should be taken in the same vein as ‘localic’ in ‘localic topos’ or ‘topological’ in ‘topological convergence’. These two terms also suggest that the study of other locales is not part of topology or that these other locales are not spaces, which is also incorrect.

The really clear term for a topological locale is ‘locale with enough points to separate the opens’, but ‘locale with enough points’ should be unambiguous. However, it is still a bit long. The shortest term, ‘spatial locale’, is probably also the most common. Occasionally one sees ‘spacial’ instead of ‘spatial’, but this might just be a misspelling.

Related concepts

• locale

• topological space

• ionad

• topos