What is a computable set? One may call a bounded subset of the plane computable if it can be drawn at any resolution on a computer screen. Using the constructive approach to computability one naturally considers totally bounded subsets of the plane. We connect this notion with notions introduced in other frameworks. A subset of a totally bounded set is again totally bounded iff it is located. L...

We describe LOCALE, a system that allows cooperating information systems to share labels for photographs. Participating photographs are enhanced with a geographic location stamp – the latitude and longitude where the photograph was taken. For a photograph with no label, LOCALE can use the shared information to assign a label based on other photographs that were taken in the same area. LOCALE th...

These theories introduce a new command called statespace. It’s usage is similar to records. However, the command does not introduce a new type but an abstract specification based on the locale infrastructure. This leads to extra flexibility in composing state space components, in particular multiple inheritance and renaming of components. The state space infrastructure basically manages the fol...

This paper analyses the notion of a positivity relation of Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finall...

There are several different standard conventions for specifying a locale, and the first decision is which convention to use. PhET specifies a locale using codes defined by the International Standards Organization (ISO). ISO 639-1 defines language codes [3], while ISO 3166-1 defines country codes [4]. While there are other standards, these are the most commonly-used definitions. An example of a ...

Searching for an object model is considered to be one of the most desirable and yet difficult searches. The problem is made difficult by the presence of clutter in a scene, as well as the fact that objects may be imaged under different lighting conditions. We have developed a feature localization scheme that finds a set of locales in an image. Our object search method matches image locales with...

This paper investigates aspects of measure and randomness in the context of locale theory (point-free topology). We prove that every measure (σ-continuous valuation) μ, on the σ-frame of opens of a fitted σ-locale X, extends to a measure on the lattice of all σ-sublocales of X (Theorem 1). Furthermore, when μ is a finite measure with μ(X) = M , the σ-locale X has a smallest σ-sublocale of measu...

It is a classic result in lattice theory that a poset is a complete lattice i↵ it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting algebra) i↵ it can be realized as fixpoints of a nucleus on the locale of upsets of a poset. He also showed how to generate a nucleus on upsets by adding a structure of “paths” to a...

We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and mostly consist of explicit manipulations on a distributive lattice associated to a given Riesz space.