If X is a locale, then its double powerlocale PX is defined to be PU(PL(X)) where PU and PL are the upper and lower powerlocale constructions. We prove various results relating it to exponentiation of locales, including the following. First, if X is a locale for which the exponential S exists (where S is the Sierpinski locale), then PX is an exponential S X . Second, if in addition W is a local...

causality; concurrency; event structures; frames; locale theory; geometric logic The conventional approach to causality is based on partial orders. Without additional structure, partial orders are only capable of expressing AND causality. In this paper we investigate a syntactic, or logical, approach to causality which allows other causal relationships, such as OR causality, to be expressed wit...

Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected. Bishop defines a metric space to be compact if it is complete and totally bounded. A subset of a totally bounded set...

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 present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform subl...

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...