An intuitionistic analysis of the relationship between pointfree and pointwise topology brings new notions to light that are hidden from a classical viewpoint. In this paper, we introduce one of these, namely the notion of reducibility for a pointfree topology, which is classically equivalent to spatiality. We study its basic properties and we relate it to spatiality and to other concepts in co...

We will present a formalization of pointfree topology in Martin-Löf's type theory. A notion of point will be introduced and we will show that the points of a Scott topology form a Scott domain. This work follows closely the intuitionistic approach to pointfree topology and domain theory, developed mainly by Martin-Löf and Sambin. The important di erence is that the de nitions and proofs are mac...

in this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. we study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. for strongly z-ideals, we analyze prime ideals using the concept of zero sets. moreover, it is proven that the intersection of all zero sets of a prime ideal of c(l),...

Abstract: The concepts of upper and lower semicontinuity in pointfree topology were introduced and first studied by Li and Wang in 1997. However Li and Wang’s treatment does not faithfully reflect the original classical notion. In this note, we present algebraic descriptions of upper and lower semicontinuous real functions, in terms of frame homomorphisms, that suggest the right alternative to ...

Pointfree topology is, as the name suggests, a way of studying spaces without (mentioning) points. This idea is more natural than one might initially think. For example, when drawing a point on paper, we do no draw an actual point, but a collection of points somewhere near the desired one. We drew a “spot”, which can be reduced in size if that would be required to serve our purposes. Hence it m...