In this paper, we introduce and study a new topology related to a self mapping on a nonempty set.Let X be a nonempty set and let f be a self mapping on X. Then the set of all invariant subsets ofX related to f, i.e. f := fA X : f(A) Ag P(X) is a topology on X. Among other things,we nd the smallest open sets contains a point x 2 X. Moreover, we find the relations between fand To f . For insta...

In a classic paper, Smirnov [14] characterized the poset of compactifications of a completely regular space in terms of the proximities on the space. Later, Smyth [15] introduced the notion of a stable compactification of a T0-space and described them in terms of quasi-proximities on the space. Banaschewski [1] formulated Smirnov’s results in the pointfree setting, defining a compactification o...

The purpose of this note is to compare the rings of continuous functions, integer-valued or real-valued, in pointfree topology with those in classical topology. To this end, it first characterizes the Boolean frames (= complete Boolean algebras) whose function rings are isomorphic to a classical one and then employs this to exhibit a large class of frames for which the functions rings are not o...

Hedin, A. 2011. Contributions to Pointfree Topology and Apartness Spaces. Department of Mathematics. Uppsala Dissertations in Mathematics 71. 40 pp. Uppsala. The work in this thesis contains some contributions to constructive point-free topology and the theory of apartness spaces. The first two papers deal with constructive domain theory using formal topology. In Paper I we focus on the notion ...

A σ-frame is a poset with countable joins and finite meets in which binary meets distribute over countable joins. Aim of this paper is to present σ-frames, actually σ-locales, as a branch of Formal Topology, that is, intuitionistic and predicative pointfree topology. We show that every σframe L can be presented as the lattice of Lindelöf elements (those whose covers admit countable subcovers) o...