nLab spatial locale

Idea

A 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 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 spatial, as defined above.
2. $L$ has enough points, as defined above.
3. Given any two opens $U$ and $V$ in $L$, $U \leq V$ if (hence iff) every point of $L$ that belongs to $U$ also belongs to $V$.
4. $L$ is isomorphic to $\Omega(pt(L))$, where $pt(L)$ is the space of points? of $L$.
5. The natural morphism $\eta_L\colon \Omega(pt(L)) \to L$ (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 spatial locales comprise the image of this adjunction. The corresponding condition on topological spaces is being sober.

Therefore, the full subcategory of $Loc$ on the spatial locales is equivalent to the full subcategory of $Top$ on sober spaces.

Terminology

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

The really clear term for a spatial 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. Occasionally one sees ‘spacial’ instead of ‘spatial’.

Criteria for spatiality

1. Hofmann–Lawson duality (1978; see Picado–Pultr, Theorem 6.4.3 and Proposition 6.3.3): assuming the axiom of choice, locally compact locales are spatial. In particular, compact regular locales are locally compact, hence automatically spatial.

2. Every coherent locale is spatial. In particular, Stone locales, Stonean locales, and hyperstonean locales are spatial.

3. The meet of a countable family of open sublocales (i.e., a $G_\delta$-sublocale) of a compact regular locale is spatial.

4. The completion of a uniform locale with a countable basis of uniformity is spatial.

Related concepts

• spatial topos, localic topos

• locale

• topological space

• ionad

• topos

References

• Jorge Picado, Aleš Pultr, Frames and Locales. Topology without points, Frontiers in Mathematics, Birkhäuser (2012).